[{"data":1,"prerenderedAt":6354},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic":3},{"post":4,"nextPost":343,"prevPost":6305},{"id":5,"title":6,"body":7,"description":322,"draft":330,"enableComment":330,"extension":331,"image":322,"important":330,"location":332,"meta":333,"navigation":334,"ogImage":332,"onday":335,"path":336,"seo":337,"stem":338,"summary":339,"tags":340,"__hash__":342},"blog\u002Fblog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic.md","The Messianic Narrative of Anthropic",{"type":8,"value":9,"toc":321},"minimark",[10,15,19,22,28,31,37,40,45,52,56,64,75,80,87,90,93,96,118,121,138,141,145,150,161,164,171,174,177,181,184,194,205,210,217,220,226,243,250,253,260,266,269,273,276,290,293,296,302,305,308,311,316],[11,12,14],"h2",{"id":13},"judaism-and-the-messiah","Judaism and the Messiah",[16,17,18],"p",{},"Within the theological framework of Judaism, the figure of the Messiah represents a systematic redemptive expectation gradually crystallized by the ancient Jewish people through prolonged suffering, exile, and doctrinal reflection. Its core impetus derives from a structural predicament: following the successive failures of monarchical authority and the Temple, the nation required a force transcending empirical reality to sustain identity and hope. After losing sovereign kingship, Temple sovereignty, and political freedom, a narrative capable of indefinitely deferred verification — yet perpetually furnishing meaning for action and ultimate hope — became an existential necessity. The Messiah constitutes the personified condensation of this narrative, bearing the weight of every unmet worldly aspiration: national independence, social justice, bodily resurrection, perpetual peace among nations, and the direct rule of the Divine.",[16,20,21],{},"Successive eras and communities have projected disparate contents into this \"formula\": the oppressed inserted armed liberation; the devout inserted Temple restoration; the afflicted inserted eschatological recompense; the guilt-ridden inserted vicarious atonement. Christianity elected the genetic strands of the \"Suffering Servant\" and the \"Heavenly Son of Man\" to define Jesus; mainstream Judaism, by contrast, ultimately rejected Jesus and preserved the yet-to-arrive, certain-to-come, thoroughly earthly Davidic Messiah.",[16,23,24],{},[25,26,27],"strong",{},"Such is the genesis of the Messianic archetype within Judaism.",[16,29,30],{},"From the perspectives of comparative mythology and political psychology, this structure approximates what may be termed an archetypal narrative in human culture:",[32,33,34],"blockquote",{},[16,35,36],{},"The world descends into crisis;\nThe masses remain unaware;\nA minority attains true knowledge;\nFrom this minority a prophet emerges;\nThe prophet suffers doubt, incomprehension, and attack;\nThe prophet, having endured tribulation, ultimately acquires authority or influence;\nSalvation is delivered to the multitude, or catastrophe averted.",[16,38,39],{},"Viewed through this lens, the Messiah transcends the boundaries of a merely religious concept; it also constitutes one of the most ancient and influential archetypes in the human narrative repertoire:",[32,41,42],{},[16,43,44],{},"When the world descends into crisis, a chosen figure emerges, bearing a mission surpassing ordinary human capacity, ultimately delivering redemption and renewal.",[16,46,47],{},[48,49],"img",{"alt":50,"src":51},"Anthropic refused to sign an initiative recognizing open-weight AI models, deliberately maintaining distance and \"independence\" from other AI companies","https:\u002F\u002Fimage-assets.dreams.plus\u002F202607310100531.png",[11,53,55],{"id":54},"the-biographical-background-of-the-amodei-siblings","The Biographical Background of the Amodei Siblings",[16,57,58,59,63],{},"The very name of the company is revealing in this regard: \"Anthropic,\" etymologically derived from the Greek ",[60,61,62],"em",{},"anthrōpos"," — \"human being\" — signals their self-conception as a technology enterprise oriented toward human-centeredness and the destiny and welfare of all humankind. This, precisely, is the role they have long labored to inhabit.",[16,65,66,67,74],{},"It is by no means coincidental that the Amodei siblings' mother was a devout Jewish adherent. Their mother, ",[68,69,73],"a",{"href":70,"rel":71},"https:\u002F\u002Fgrokipedia.com\u002Fpage\u002FElena_Engel#ref-19",[72],"nofollow","Elena Engel",", is a Jewish-American library project manager; their father, Riccardo Amodei, was an Italian leather artisan who suffered chronic illness and passed away when Dario was still young. The present author submits that there are reasonable grounds to suspect a correlation between Dario's preoccupation with the Messianic narrative and the circumstances of his familial upbringing. In Jewish-American households, domestic instruction frequently occupies a formative role.",[32,76,77],{},[16,78,79],{},"As a mother, Engel played a key role in fostering her children's intellectual curiosity and ethical outlook, instilling a strong sense of right and wrong as well as responsibility toward improving the world.",[16,81,82,83,86],{},"This educational paradigm, which internalizes the repair of the world (",[60,84,85],{},"Tikkun Olam",", a core ethical concept in Judaism) as an individual vocation, is in its essence a secularized Messianic spirit: the conviction that human action, rather than pure divine intervention, can propel the world toward perfection. The premature death of the father and the absence of a stable masculine pillar in the developmental environment may have further intensified Dario's psychological need for a grand, determinate, and ultimate explanatory framework.",[16,88,89],{},"The Messianic narrative supplies precisely such a structure: it promises a definitive inflection point from chaos to order, from suffering to redemption. His technological optimism — and indeed his faith in the salvific potential of artificial general intelligence — may thus be understood not as a deduction from pure rationality, but as the projection, onto the technological age, of an ethical-redemptive complex molded by family.",[16,91,92],{},"Why, in the era of AGI, would a technology company so spontaneously adopt discursive structures that historically belonged to the domains of religion, philosophy of history, and eschatology?",[16,94,95],{},"The present author contends that traditional religion and eschatology principally address three questions:",[97,98,99,106,112],"ul",{},[100,101,102,105],"li",{},[25,103,104],{},"Explaining everything",": Why is the world as it is? What are the root causes of suffering, chaos, and injustice?",[100,107,108,111],{},[25,109,110],{},"Promising salvation",": What constitutes the ultimate, perfect future state? (e.g., the Millennium, Heaven, the earthly Kingdom of God)",[100,113,114,117],{},[25,115,116],{},"Indicating the path",": Through what means (faith, spiritual practice, revolution) can that future be attained?",[16,119,120],{},"With the emergence of AGI, these three dimensions find precise correspondence:",[97,122,123,128,133],{},[100,124,125,127],{},[25,126,104],{},": The root of all human afflictions — disease, poverty, conflict, even mortality itself — lies in \"insufficient intelligence.\" Extant human cognition and conventional computation cannot process information of adequate complexity.",[100,129,130,132],{},[25,131,110],{},": AGI or ASI (artificial superintelligence) will constitute an omniscient, omnibenevolent, quasi-omnipotent \"godlike entity,\" resolving energy crises, climate change, and disease, and even enabling \"mind uploading\" and \"digital immortality.\"",[100,134,135,137],{},[25,136,116],{},": The technological trajectory — computational scaling, algorithmic breakthroughs, alignment research.",[16,139,140],{},"To be clear, we cannot, on this basis, assert that Anthropic's value system derives directly from familial background, nor can we ascertain whether it bears any relationship to Israel; nor can we simply reduce its AI safety philosophy to personal developmental history. Nevertheless, this formative milieu may provide a useful interpretive lens for understanding why Dario has long gravitated toward apocalyptic and safety-centric narratives.",[11,142,144],{"id":143},"the-narrative-of-capital-apocalyptic-marketing","The Narrative of Capital: Apocalyptic Marketing",[32,146,147],{},[16,148,149],{},"AI is perilous; only we can save AI.",[16,151,152,153,156,157,160],{},"In the modern commercial system, the dynamic behavior of markets is not determined solely by fundamental variables but is, to a substantial degree, governed by two mutually interwoven psychological variables: ",[60,154,155],{},"expectation"," and ",[60,158,159],{},"sentiment",".",[16,162,163],{},"Expectation constitutes the rational anchor of capital allocation behavior — it is grounded in deductions concerning future cash flows, technological trajectories, or policy environments, forming a logical framework subject to periodic verification. Sentiment, by contrast, functions as the nonlinear accelerator of expectation realization: driven by collective psychological states such as greed and fear, it amplifies or distorts information, thereby engendering systematic deviations of price from intrinsic value.",[16,165,166,167,170],{},"It is worth noting that capital is, in its essence, ",[60,168,169],{},"narrative-preferring",": it seeks out and rewards stories that simultaneously exhibit simplicity, grand vision, and verifiable milestones. Only such narratives can efficiently aggregate consensus within an environment of information asymmetry, converting dispersed individual beliefs into coherent asset-pricing behavior.",[16,172,173],{},"Put differently, commercial competition has, to a considerable extent, evolved into a contest of competence in managing expectations and steering sentiment. The former demands that enterprises continually honor their narrative commitments through clear milestones; the latter demands prudent calibration in response to excessive fluctuations in market psychology.",[16,175,176],{},"Whether a commercial narrative can sustain capital attraction, then, is determined not by its fantastical appeal, but by whether it can, across the dual dimensions of expectation and sentiment, both ignite the imagination and withstand the scrutiny of periodic factual verification.",[11,178,180],{"id":179},"malefactors-and-prophets","Malefactors and Prophets",[16,182,183],{},"Against the backdrop of intensifying Sino-American antagonism, Anthropic has exploited a Cold War stereotype deeply embedded in modern Western discourse: the portrayal of China as an authoritarian, malevolent, despotic, and formidable power — a threat not merely to its own populace but to the entire democratic world. Anthropic, cast in the role of the minority \"prophet\" who has discerned the danger ahead of others, must take action; the \"democratic camp\" must retain AI supremacy and assume the mantle of guardianship. The company has thus fashioned itself as a democratic, progressive, safety-oriented AI enterprise.",[16,185,186,187,160],{},"In recent times, Anthropic has not spared its criticism of Chinese large language models, alleging that they have performed distillation upon its Claude series of models. This is documented in the February 2026 report ",[60,188,189],{},[68,190,193],{"href":191,"rel":192},"https:\u002F\u002Fwww.anthropic.com\u002Fnews\u002Fdetecting-and-preventing-distillation-attacks",[72],"Detecting and Preventing Distillation Attacks",[16,195,196,197,204],{},"Implicit within this construction is a metaphor: a regime of the \"authoritarian and malevolent\" variety, such as China, allegedly lacks vigilance and rationality regarding AI, and will abuse or recklessly permit AI technologies to inflict harm. Following President Trump's visit to China in May 2026, Anthropic published a report titled ",[60,198,199],{},[68,200,203],{"href":201,"rel":202},"https:\u002F\u002Fwww.anthropic.com\u002Fresearch\u002F2028-ai-leadership",[72],"2028: Two Scenarios for Global AI Leadership",", in which it stated:",[32,206,207],{},[16,208,209],{},"If the frontier is set by regimes that treat AI as an instrument of repression, military advantage over democracies, and domestic control, the transition is less likely to go well, for those regimes' own citizens or anyone else.\nHistorically, the reach of authoritarian rule has been limited by its dependence on human enforcers to carry out surveillance and repression. Powerful AI systems may remove that dependency, enabling automated repression on a far greater scale. For that reason, the prospect of the CCP leading in AI is among the greatest threats to a successful transition.",[16,211,212,213,216],{},"From this logic, the Chinese Communist Party is no longer merely a state regime but is recast as a ",[60,214,215],{},"civilizational risk",". This, in fact, already exceeds the traditional narrative of interstate competition, and it is here that Anthropic's approach becomes most peculiar. For ordinary interstate competition is typically framed around American interests versus Chinese interests. Anthropic's narrative, however, operates on an elevated discursive plane: the interests of the free world versus the risk to humanity's shared future.",[16,218,219],{},"Herein lies an \"eschatological escalation.\" In the preceding century, the Cold War narrative held that a Soviet victory would threaten the \"free world.\" In the AI era, the narrative has mutated into: the triumph of the wrong power → the endangerment of the future of human civilization.",[16,221,222,225],{},[25,223,224],{},"AI has been invested with a status approaching that of a savior — or an apocalypse."," If AGI truly is what its proponents believe it to be — surpassing all humans in intelligence, commanding scientific research, commanding the economy, commanding military affairs — then whoever first possesses AGI no longer merely possesses a technology, but rather the capacity to define the civilization of the future. AI safety, national security, and civilizational security are thereby progressively collapsed into a single question.",[16,227,228,229,232,233,232,236,232,239,242],{},"AI is, moreover, naturally suited to bear this narrative, insofar as it simultaneously possesses four amplifiers: immense unknowns, world-transforming potential, apocalyptic risk scenarios, and soteriological technological vision. One consequently observes, throughout the Anthropic community and Dario's blog, an enthusiasm for discussing AI in language approaching the theological: ",[60,230,231],{},"Superintelligence",", ",[60,234,235],{},"Alignment",[60,237,238],{},"the Control Problem",[60,240,241],{},"Existential Risk",". While these concepts carry technical significance, they are, at the cultural level, readily mapped onto the traditional eschatological framework.",[16,244,245,246,249],{},"In religious allegory, the chain of ",[60,247,248],{},"crisis → enemy → guardian → redemptive solution"," constitutes an ancient and highly efficient mode of propagation. From this vantage point, AI companies may not merely be discussing technical problems; they may also be participating in a larger political and religious narrative.",[16,251,252],{},"Viewed accordingly, the competition among AI companies is not only a contest of model capabilities but also a struggle for discursive authority: who is qualified to define the risks of the future, who is qualified to play the guardian, and who is qualified to furnish the path to redemption.",[16,254,255,256,259],{},"However, the present author submits that what warrants the greatest vigilance here is not conspiracy but ",[60,257,258],{},"sacralization",". The critical point is not whether Anthropic deliberately exploits Cold War stereotypes. What merits closer attention is that when a single organization simultaneously holds the authority to define risk, to interpret technology, and to interpret morality, it readily becomes invested with a certain \"sacredness.\" And the catastrophic mass movements of history have, more often than not, been carried out precisely within the framework of such \"sacralizing narratives.\"",[16,261,262,265],{},[60,263,264],{},"We know what the risk is; we know what the future is; we know what the correct path is."," This structure is a frequent guest in the histories of religious organizations, revolutionary parties, and state apparatuses. But for it to appear within a major technology company, elevated to the status of core values, is comparatively rare.",[16,267,268],{},"Historical experience instructs us that any force claiming to have simultaneously grasped \"the future,\" \"the truth,\" and \"moral legitimacy\" ought to be subjected to sustained scrutiny and oversight.",[11,270,272],{"id":271},"conclusion","Conclusion",[16,274,275],{},"If one could still debate whether Anthropic's earlier posturing toward China and the U.S. Department of Defense amounted to \"brand theater,\" the present author maintains that the releases of Mythos and Fable represent the eve of Anthropic's narrative bankruptcy.",[16,277,278,279,282,283,286,287,160],{},"From the foregoing analysis, it becomes apparent that the predominantly negative online reception of Anthropic may, in fact, serve its purposes rather well. A degree of negative appraisal is not necessarily a loss; it may instead reinforce the company's distinctive positioning. Anthropic is quite content to style itself as a \"prophet\" or \"Messiah\"-type figure. It is not merely selling a model; it is also selling an ",[60,280,281],{},"identity"," — a unique symbolic marker within the capital market. Anthropic's conduct has, throughout, been reiterating a single refrain: ",[60,284,285],{},"we are not like the other AI companies."," When a technology becomes sufficiently consequential, what companies compete on is no longer solely the product but rather the ",[60,288,289],{},"civilizational role",[16,291,292],{},"More intriguingly, this touches upon an economic question: in the AI era, \"safety\" is itself being transformed into a form of scarce brand equity.",[16,294,295],{},"If everyone can train models, then \"who is fastest,\" \"who is cheapest,\" and \"who is safest\" all become dimensions of competition — and Anthropic manifestly seeks to occupy the third. In its imagined division of civilizational labor within the capital market, OpenAI is the innovator, Google is the technology titan, Meta is the open-source champion, and Anthropic is the guardian — all facing a common adversary: \"malevolent China.\" And once the discourse enters the register of \"who is qualified to define the risks of the future and who is qualified to represent the interests of humanity,\" the debate naturally acquires theological and political-theological inflections, ceasing to be a merely engineering matter.",[16,297,298,299,160],{},"Whether or not one endorses this image, it indisputably possesses strong market differentiation, and the narrative is undeniably compelling. It must, at this juncture, be conceded that Anthropic is indeed adept at ",[60,300,301],{},"storytelling",[16,303,304],{},"Nevertheless, the present author retains reservations regarding the story Anthropic tells, and the sustainability of this narrative remains open to question. For under a liberal market regime, in no other domain, in no other historical period, in no other market, has there ever been a precedent of capital willingly accepting compromise with \"safety\" as the selling point. Capital is inherently profit-seeking; where the returns are sufficiently high, manufacturing and selling the rope with which one is to be hanged is by no means an uncommon occurrence. The so-called \"guardian narrative\" is perhaps more a matter of brand positioning than a principle for which one is genuinely prepared to bear costs. From the standpoint of political economy, what ultimately determines the strength of an organization's convictions is seldom what it writes in its blog posts, but rather what it sacrifices when interests and principles come into conflict.",[16,306,307],{},"The present author holds that, from a Chinese perspective, there is no need whatsoever to engage with Anthropic's theatrics. If one strips away Anthropic's narrative, examining it purely from the perspective of corporate behavior, it is, regardless of the narrative framework it deploys, fundamentally a commercial enterprise. Commerce is inseparable from customers, financing, and IPO listings. It cannot wholly detach itself from commercial interests.",[16,309,310],{},"From the perspective of American strategic circles, the core anxiety is not China's ethnic character but rather the prospect that a political system divergent from the liberal democratic tradition might acquire the capacity to define the technical rules of the future. If China were to lead in the AGI competition, the future world order might be shaped according to Chinese institutional preferences. What is truly feared here is not ethnic identity but institutional competition.",[16,312,313],{},[25,314,315],{},"In brief: for the United States and the West, heirs to over a century of liberal tradition, the prospect of a political regime at variance with the liberal-democratic value system acquiring the capacity to define the technical rules of the future is entirely unacceptable. It is precisely this anxiety that furnishes Anthropic with the operational space to fish in troubled waters.",[16,317,318],{},[60,319,320],{},"Fin.",{"title":322,"searchDepth":323,"depth":323,"links":324},"",2,[325,326,327,328,329],{"id":13,"depth":323,"text":14},{"id":54,"depth":323,"text":55},{"id":143,"depth":323,"text":144},{"id":179,"depth":323,"text":180},{"id":271,"depth":323,"text":272},false,"md",null,{},true,"2026-05-26","\u002Fblog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic",{"title":6,"description":322},"blog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic","The theatrics of Anthropic, led by Dario Amodei, exhibit the quintessential characteristics of the Judaic Messianic narrative, whose structure typically proceeds as follows: the world confronts an immense crisis of which the majority remains unaware; only a select few prophets perceive the danger; these prophets must be entrusted with authority; and they shall deliver salvation to the world.",[341],"thoughts","9nTzPPP27OlwW56L0Y8-Mp5Hwd0G2aEZxGbeleyFPjM",{"id":344,"title":345,"body":346,"description":322,"draft":330,"enableComment":334,"extension":331,"image":322,"important":330,"location":332,"meta":6295,"navigation":334,"ogImage":332,"onday":6296,"path":6297,"seo":6298,"stem":6299,"summary":6300,"tags":6301,"__hash__":6304},"blog\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor.md","张量 Tensor 简述",{"type":8,"value":347,"toc":6284},[348,353,356,359,362,365,368,371,383,390,442,445,448,451,737,740,1094,1097,1402,1405,1445,1486,1644,1647,1650,1653,1656,1806,1809,2294,2297,2305,2308,2393,2396,2399,2402,2501,2504,2507,2620,2628,2780,2846,2849,2910,3017,3127,3235,3238,3241,3332,3335,3517,3520,3687,3742,3745,3748,3898,5545,5548,5856,5859,5862,5865,6251,6254,6257,6269,6272,6275,6278,6281],[32,349,350],{},[16,351,352],{},"张量是标量、向量和矩阵的高维推广",[16,354,355],{},"你几乎肯定见过这句话。它出现在几乎所有教科书的第一页，也几乎是所有困惑的出发点。",[16,357,358],{},"原因在于，这句话只描述了张量的存储格式，而对它的本质只字未提。这就好比说「人是一堆细胞的集合」——不算错，却没有回答「人到底是什么」这个真正的问题。",[16,360,361],{},"本文只追求一个目标：为什么物理学家和数学家如此坚持区分向量与「对偶向量」，并坚持区分张量中的「上标」与「下标」？只要你能抓住这一点，围绕张量的所有神秘感都会消散。",[16,363,364],{},"我们按以下顺序展开：先重新审视向量本身；接着介绍它的孪生兄弟——对偶向量；然后给出张量的严格定义；用一个具体例子拆解一个常见的困惑点；最后阐明连接向量与对偶向量的那座关键桥梁——度量。",[11,366,367],{"id":367},"箭头",[16,369,370],{},"假设平面上有一支物理意义上的箭头，指向北偏东 30°，长 5 米。这支箭头具有客观存在性——无论你如何建立坐标系，它都在原处；它的长度和方向并不会因为你的参考系变了就改变。",[16,372,373,374,378,379,382],{},"然而，当你写下 ",[375,376,377],"code",{},"(3, 4)"," 这样的坐标时，你已经做了一次隐蔽的操作：你选择了一个坐标基，然后把这个箭头相对于该基「翻译」成了一组数字。换一个基（比如把坐标轴旋转 45°），同一支箭头会被翻译成完全不同的数字，比如 ",[375,380,381],{},"(4.95, 0.71)","。",[16,384,385,386,389],{},"数字变了，箭头没变。这正是向量作为几何对象与其坐标表示（数组）之间最根本的区别。深度学习里的张量（例如 ",[375,387,388],{},"torch.tensor([3,4])","）只是这个「翻译结果」，从不关心背后是否站着一个不变的几何对象。纯粹做数值计算时这完全没问题，但一旦你想谈论「物理定律不依赖于观察者的选择」这类命题，就必须回到箭头本身。",[16,391,392,393,441],{},"抛开坐标不谈，向量空间 ",[394,395,399,421],"span",{"className":396,"translate":398},[397],"katex","no",[394,400,403],{"className":401},[402],"katex-mathml",[404,405,407],"math",{"xmlns":406},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[408,409,410,417],"semantics",{},[411,412,413],"mrow",{},[414,415,416],"mi",{},"V",[418,419,416],"annotation",{"encoding":420},"application\u002Fx-tex",[394,422,426],{"className":423,"ariaHidden":425},[424],"katex-html","true",[394,427,430,435],{"className":428},[429],"base",[394,431],{"className":432,"style":434},[433],"strut","height:0.6833em;",[394,436,416],{"className":437,"style":440},[438,439],"mord","mathnormal","margin-right:0.2222em;"," 的定义其实相当简洁：只要一个对象集合支持加法（首尾相接的合成）和标量乘法（拉伸或压缩），并满足少数几条自然的代数定律，它就构成一个向量空间，其元素就是向量。速度、力、位移都是这类对象。",[11,443,444],{"id":444},"对偶空间",[16,446,447],{},"向量是箭头；那么，「测量」这支箭头的工具又是什么？",[16,449,450],{},"它是一种完全不同类型的对象，叫作对偶向量（也叫余向量或 1-形式）。它不是箭头，而是一把标尺——一组等距的平行线。把箭头放上去，箭头穿过的线数就是它输出的数字。",[16,452,453,454,529,530,558,559,617,618,650,651,685,686,736],{},"严格定义：对偶空间 ",[394,455,457,478],{"className":456,"translate":398},[397],[394,458,460],{"className":459},[402],[404,461,462],{"xmlns":406},[408,463,464,475],{},[411,465,466],{},[467,468,469,471],"msup",{},[414,470,416],{},[472,473,474],"mo",{},"∗",[418,476,477],{"encoding":420},"V^*",[394,479,481],{"className":480,"ariaHidden":425},[424],[394,482,484,488],{"className":483},[429],[394,485],{"className":486,"style":487},[433],"height:0.6887em;",[394,489,491,494],{"className":490},[438],[394,492,416],{"className":493,"style":440},[438,439],[394,495,498],{"className":496},[497],"msupsub",[394,499,502],{"className":500},[501],"vlist-t",[394,503,506],{"className":504},[505],"vlist-r",[394,507,510],{"className":508,"style":487},[509],"vlist",[394,511,513,518],{"style":512},"top:-3.063em;margin-right:0.05em;",[394,514],{"className":515,"style":517},[516],"pstrut","height:2.7em;",[394,519,525],{"className":520},[521,522,523,524],"sizing","reset-size6","size3","mtight",[394,526,474],{"className":527},[528,524],"mbin"," 是 ",[394,531,533,546],{"className":532,"translate":398},[397],[394,534,536],{"className":535},[402],[404,537,538],{"xmlns":406},[408,539,540,544],{},[411,541,542],{},[414,543,416],{},[418,545,416],{"encoding":420},[394,547,549],{"className":548,"ariaHidden":425},[424],[394,550,552,555],{"className":551},[429],[394,553],{"className":554,"style":434},[433],[394,556,416],{"className":557,"style":440},[438,439]," 上所有线性泛函的集合。也就是说，",[394,560,562,579],{"className":561,"translate":398},[397],[394,563,565],{"className":564},[402],[404,566,567],{"xmlns":406},[408,568,569,577],{},[411,570,571],{},[467,572,573,575],{},[414,574,416],{},[472,576,474],{},[418,578,477],{"encoding":420},[394,580,582],{"className":581,"ariaHidden":425},[424],[394,583,585,588],{"className":584},[429],[394,586],{"className":587,"style":487},[433],[394,589,591,594],{"className":590},[438],[394,592,416],{"className":593,"style":440},[438,439],[394,595,597],{"className":596},[497],[394,598,600],{"className":599},[501],[394,601,603],{"className":602},[505],[394,604,606],{"className":605,"style":487},[509],[394,607,608,611],{"style":512},[394,609],{"className":610,"style":517},[516],[394,612,614],{"className":613},[521,522,523,524],[394,615,474],{"className":616},[528,524]," 中的每个元素 ",[394,619,621,636],{"className":620,"translate":398},[397],[394,622,624],{"className":623},[402],[404,625,626],{"xmlns":406},[408,627,628,633],{},[411,629,630],{},[414,631,632],{},"ω",[418,634,635],{"encoding":420},"\\omega",[394,637,639],{"className":638,"ariaHidden":425},[424],[394,640,642,646],{"className":641},[429],[394,643],{"className":644,"style":645},[433],"height:0.4306em;",[394,647,632],{"className":648,"style":649},[438,439],"margin-right:0.0359em;"," 都是一条规则：给它一个向量 ",[394,652,654,670],{"className":653,"translate":398},[397],[394,655,657],{"className":656},[402],[404,658,659],{"xmlns":406},[408,660,661,667],{},[411,662,663],{},[414,664,666],{"mathvariant":665},"bold","v",[418,668,669],{"encoding":420},"\\mathbf{v}",[394,671,673],{"className":672,"ariaHidden":425},[424],[394,674,676,680],{"className":675},[429],[394,677],{"className":678,"style":679},[433],"height:0.4444em;",[394,681,666],{"className":682,"style":684},[438,683],"mathbf","margin-right:0.016em;","，它输出一个实数 ",[394,687,689,712],{"className":688,"translate":398},[397],[394,690,692],{"className":691},[402],[404,693,694],{"xmlns":406},[408,695,696,709],{},[411,697,698,700,704,706],{},[414,699,632],{},[472,701,703],{"stretchy":702},"false","(",[414,705,666],{"mathvariant":665},[472,707,708],{"stretchy":702},")",[418,710,711],{"encoding":420},"\\omega(\\mathbf{v})",[394,713,715],{"className":714,"ariaHidden":425},[424],[394,716,718,722,725,729,732],{"className":717},[429],[394,719],{"className":720,"style":721},[433],"height:1em;vertical-align:-0.25em;",[394,723,632],{"className":724,"style":649},[438,439],[394,726,703],{"className":727},[728],"mopen",[394,730,666],{"className":731,"style":684},[438,683],[394,733,708],{"className":734},[735],"mclose","，并且这条规则对向量加法和标量乘法都是线性的。",[16,738,739],{},"在有限维线性代数中，这简单到几乎平庸：",[97,741,742,913],{},[100,743,744,745,773,774],{},"向量 ",[394,746,748,761],{"className":747,"translate":398},[397],[394,749,751],{"className":750},[402],[404,752,753],{"xmlns":406},[408,754,755,759],{},[411,756,757],{},[414,758,666],{"mathvariant":665},[418,760,669],{"encoding":420},[394,762,764],{"className":763,"ariaHidden":425},[424],[394,765,767,770],{"className":766},[429],[394,768],{"className":769,"style":679},[433],[394,771,666],{"className":772,"style":684},[438,683]," 是列向量：",[394,775,777,824],{"className":776,"translate":398},[397],[394,778,780],{"className":779},[402],[404,781,782],{"xmlns":406},[408,783,784,821],{},[411,785,786,789,818],{},[472,787,788],{"fence":425},"[",[790,791,795,809],"mtable",{"rowspacing":792,"columnalign":793,"columnspacing":794},"0.16em","center","1em",[796,797,798],"mtr",{},[799,800,801],"mtd",{},[802,803,805],"mstyle",{"scriptlevel":804,"displaystyle":702},"0",[806,807,808],"mn",{},"2",[796,810,811],{},[799,812,813],{},[802,814,815],{"scriptlevel":804,"displaystyle":702},[806,816,817],{},"3",[472,819,820],{"fence":425},"]",[418,822,823],{"encoding":420},"\\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix}",[394,825,827],{"className":826,"ariaHidden":425},[424],[394,828,830,834],{"className":829},[429],[394,831],{"className":832,"style":833},[433],"height:2.4em;vertical-align:-0.95em;",[394,835,838,847,907],{"className":836},[837],"minner",[394,839,843],{"className":840,"style":842},[728,841],"delimcenter","top:0em;",[394,844,788],{"className":845},[846,523],"delimsizing",[394,848,850],{"className":849},[438],[394,851,853],{"className":852},[790],[394,854,857],{"className":855},[856],"col-align-c",[394,858,861,898],{"className":859},[501,860],"vlist-t2",[394,862,864,893],{"className":863},[505],[394,865,868,881],{"className":866,"style":867},[509],"height:1.45em;",[394,869,871,875],{"style":870},"top:-3.61em;",[394,872],{"className":873,"style":874},[516],"height:3em;",[394,876,878],{"className":877},[438],[394,879,808],{"className":880},[438],[394,882,884,887],{"style":883},"top:-2.41em;",[394,885],{"className":886,"style":874},[516],[394,888,890],{"className":889},[438],[394,891,817],{"className":892},[438],[394,894,897],{"className":895},[896],"vlist-s","​",[394,899,901],{"className":900},[505],[394,902,905],{"className":903,"style":904},[509],"height:0.95em;",[394,906],{},[394,908,910],{"className":909,"style":842},[735,841],[394,911,820],{"className":912},[846,523],[100,914,915,916,944,945],{},"对偶向量 ",[394,917,919,932],{"className":918,"translate":398},[397],[394,920,922],{"className":921},[402],[404,923,924],{"xmlns":406},[408,925,926,930],{},[411,927,928],{},[414,929,632],{},[418,931,635],{"encoding":420},[394,933,935],{"className":934,"ariaHidden":425},[424],[394,936,938,941],{"className":937},[429],[394,939],{"className":940,"style":645},[433],[394,942,632],{"className":943,"style":649},[438,439]," 是行向量：",[394,946,948,983],{"className":947,"translate":398},[397],[394,949,951],{"className":950},[402],[404,952,953],{"xmlns":406},[408,954,955,980],{},[411,956,957,959,978],{},[472,958,788],{"fence":425},[790,960,962],{"rowspacing":792,"columnalign":961,"columnspacing":794},"center center",[796,963,964,971],{},[799,965,966],{},[802,967,968],{"scriptlevel":804,"displaystyle":702},[806,969,970],{},"1",[799,972,973],{},[802,974,975],{"scriptlevel":804,"displaystyle":702},[806,976,977],{},"4",[472,979,820],{"fence":425},[418,981,982],{"encoding":420},"\\begin{bmatrix} 1 & 4 \\end{bmatrix}",[394,984,986],{"className":985,"ariaHidden":425},[424],[394,987,989,993],{"className":988},[429],[394,990],{"className":991,"style":992},[433],"height:1.2em;vertical-align:-0.35em;",[394,994,996,1003,1088],{"className":995},[837],[394,997,999],{"className":998,"style":842},[728,841],[394,1000,788],{"className":1001},[846,1002],"size1",[394,1004,1006],{"className":1005},[438],[394,1007,1009,1046,1051,1054],{"className":1008},[790],[394,1010,1012],{"className":1011},[856],[394,1013,1015,1037],{"className":1014},[501,860],[394,1016,1018,1034],{"className":1017},[505],[394,1019,1022],{"className":1020,"style":1021},[509],"height:0.85em;",[394,1023,1025,1028],{"style":1024},"top:-3.01em;",[394,1026],{"className":1027,"style":874},[516],[394,1029,1031],{"className":1030},[438],[394,1032,970],{"className":1033},[438],[394,1035,897],{"className":1036},[896],[394,1038,1040],{"className":1039},[505],[394,1041,1044],{"className":1042,"style":1043},[509],"height:0.35em;",[394,1045],{},[394,1047],{"className":1048,"style":1050},[1049],"arraycolsep","width:0.5em;",[394,1052],{"className":1053,"style":1050},[1049],[394,1055,1057],{"className":1056},[856],[394,1058,1060,1080],{"className":1059},[501,860],[394,1061,1063,1077],{"className":1062},[505],[394,1064,1066],{"className":1065,"style":1021},[509],[394,1067,1068,1071],{"style":1024},[394,1069],{"className":1070,"style":874},[516],[394,1072,1074],{"className":1073},[438],[394,1075,977],{"className":1076},[438],[394,1078,897],{"className":1079},[896],[394,1081,1083],{"className":1082},[505],[394,1084,1086],{"className":1085,"style":1043},[509],[394,1087],{},[394,1089,1091],{"className":1090,"style":842},[735,841],[394,1092,820],{"className":1093},[846,1002],[16,1095,1096],{},"它们的「配对」就是矩阵乘法：",[394,1098,1101],{"className":1099,"translate":398},[1100],"katex-display",[394,1102,1104,1179],{"className":1103,"translate":398},[397],[394,1105,1107],{"className":1106},[402],[404,1108,1110],{"xmlns":406,"display":1109},"block",[408,1111,1112,1176],{},[411,1113,1114,1116,1118,1120,1122,1125,1147,1171,1173],{},[414,1115,632],{},[472,1117,703],{"stretchy":702},[414,1119,666],{"mathvariant":665},[472,1121,708],{"stretchy":702},[472,1123,1124],{},"=",[411,1126,1127,1129,1145],{},[472,1128,788],{"fence":425},[790,1130,1131],{"rowspacing":792,"columnalign":961,"columnspacing":794},[796,1132,1133,1139],{},[799,1134,1135],{},[802,1136,1137],{"scriptlevel":804,"displaystyle":702},[806,1138,970],{},[799,1140,1141],{},[802,1142,1143],{"scriptlevel":804,"displaystyle":702},[806,1144,977],{},[472,1146,820],{"fence":425},[411,1148,1149,1151,1169],{},[472,1150,788],{"fence":425},[790,1152,1153,1161],{"rowspacing":792,"columnalign":793,"columnspacing":794},[796,1154,1155],{},[799,1156,1157],{},[802,1158,1159],{"scriptlevel":804,"displaystyle":702},[806,1160,808],{},[796,1162,1163],{},[799,1164,1165],{},[802,1166,1167],{"scriptlevel":804,"displaystyle":702},[806,1168,817],{},[472,1170,820],{"fence":425},[472,1172,1124],{},[806,1174,1175],{},"14",[418,1177,1178],{"encoding":420},"\\omega(\\mathbf{v}) = \\begin{bmatrix} 1 & 4 \\end{bmatrix} \\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix} = 14",[394,1180,1182,1212,1392],{"className":1181,"ariaHidden":425},[424],[394,1183,1185,1188,1191,1194,1197,1200,1205,1209],{"className":1184},[429],[394,1186],{"className":1187,"style":721},[433],[394,1189,632],{"className":1190,"style":649},[438,439],[394,1192,703],{"className":1193},[728],[394,1195,666],{"className":1196,"style":684},[438,683],[394,1198,708],{"className":1199},[735],[394,1201],{"className":1202,"style":1204},[1203],"mspace","margin-right:0.2778em;",[394,1206,1124],{"className":1207},[1208],"mrel",[394,1210],{"className":1211,"style":1204},[1203],[394,1213,1215,1218,1313,1317,1383,1386,1389],{"className":1214},[429],[394,1216],{"className":1217,"style":833},[433],[394,1219,1221,1227,1307],{"className":1220},[837],[394,1222,1224],{"className":1223,"style":842},[728,841],[394,1225,788],{"className":1226},[846,1002],[394,1228,1230],{"className":1229},[438],[394,1231,1233,1267,1270,1273],{"className":1232},[790],[394,1234,1236],{"className":1235},[856],[394,1237,1239,1259],{"className":1238},[501,860],[394,1240,1242,1256],{"className":1241},[505],[394,1243,1245],{"className":1244,"style":1021},[509],[394,1246,1247,1250],{"style":1024},[394,1248],{"className":1249,"style":874},[516],[394,1251,1253],{"className":1252},[438],[394,1254,970],{"className":1255},[438],[394,1257,897],{"className":1258},[896],[394,1260,1262],{"className":1261},[505],[394,1263,1265],{"className":1264,"style":1043},[509],[394,1266],{},[394,1268],{"className":1269,"style":1050},[1049],[394,1271],{"className":1272,"style":1050},[1049],[394,1274,1276],{"className":1275},[856],[394,1277,1279,1299],{"className":1278},[501,860],[394,1280,1282,1296],{"className":1281},[505],[394,1283,1285],{"className":1284,"style":1021},[509],[394,1286,1287,1290],{"style":1024},[394,1288],{"className":1289,"style":874},[516],[394,1291,1293],{"className":1292},[438],[394,1294,977],{"className":1295},[438],[394,1297,897],{"className":1298},[896],[394,1300,1302],{"className":1301},[505],[394,1303,1305],{"className":1304,"style":1043},[509],[394,1306],{},[394,1308,1310],{"className":1309,"style":842},[735,841],[394,1311,820],{"className":1312},[846,1002],[394,1314],{"className":1315,"style":1316},[1203],"margin-right:0.1667em;",[394,1318,1320,1326,1377],{"className":1319},[837],[394,1321,1323],{"className":1322,"style":842},[728,841],[394,1324,788],{"className":1325},[846,523],[394,1327,1329],{"className":1328},[438],[394,1330,1332],{"className":1331},[790],[394,1333,1335],{"className":1334},[856],[394,1336,1338,1369],{"className":1337},[501,860],[394,1339,1341,1366],{"className":1340},[505],[394,1342,1344,1355],{"className":1343,"style":867},[509],[394,1345,1346,1349],{"style":870},[394,1347],{"className":1348,"style":874},[516],[394,1350,1352],{"className":1351},[438],[394,1353,808],{"className":1354},[438],[394,1356,1357,1360],{"style":883},[394,1358],{"className":1359,"style":874},[516],[394,1361,1363],{"className":1362},[438],[394,1364,817],{"className":1365},[438],[394,1367,897],{"className":1368},[896],[394,1370,1372],{"className":1371},[505],[394,1373,1375],{"className":1374,"style":904},[509],[394,1376],{},[394,1378,1380],{"className":1379,"style":842},[735,841],[394,1381,820],{"className":1382},[846,523],[394,1384],{"className":1385,"style":1204},[1203],[394,1387,1124],{"className":1388},[1208],[394,1390],{"className":1391,"style":1204},[1203],[394,1393,1395,1399],{"className":1394},[429],[394,1396],{"className":1397,"style":1398},[433],"height:0.6444em;",[394,1400,1175],{"className":1401},[438],[16,1403,1404],{},"所有行向量构成的空间，正是列向量空间的对偶空间。这听起来像是同一组数字换了个摆法——但当坐标系发生变换时，这两个对象的「命运」迥异，这正是问题的关键所在。",[16,1406,1407,1408,1444],{},"想象把单位从米换成厘米（这相当于把坐标基缩小为原来的 ",[394,1409,1411,1432],{"className":1410,"translate":398},[397],[394,1412,1414],{"className":1413},[402],[404,1415,1416],{"xmlns":406},[408,1417,1418,1429],{},[411,1419,1420,1422,1426],{},[806,1421,970],{},[414,1423,1425],{"mathvariant":1424},"normal","\u002F",[806,1427,1428],{},"100",[418,1430,1431],{"encoding":420},"1\u002F100",[394,1433,1435],{"className":1434,"ariaHidden":425},[424],[394,1436,1438,1441],{"className":1437},[429],[394,1439],{"className":1440,"style":721},[433],[394,1442,1431],{"className":1443},[438],"）。",[97,1446,1447,1450],{},[100,1448,1449],{},"以厘米度量时，向量分量被放大 100 倍（同一支箭头，单位更小，需要的刻度数更多）。",[100,1451,1452,1453,1485],{},"而对偶向量分量（即「每单位对应多少刻度」的标尺密度量）则缩小为 ",[394,1454,1456,1473],{"className":1455,"translate":398},[397],[394,1457,1459],{"className":1458},[402],[404,1460,1461],{"xmlns":406},[408,1462,1463,1471],{},[411,1464,1465,1467,1469],{},[806,1466,970],{},[414,1468,1425],{"mathvariant":1424},[806,1470,1428],{},[418,1472,1431],{"encoding":420},[394,1474,1476],{"className":1475,"ariaHidden":425},[424],[394,1477,1479,1482],{"className":1478},[429],[394,1480],{"className":1481,"style":721},[433],[394,1483,1431],{"className":1484},[438],"（刻度线本身变得更密；密度本身不变，但「每单位多少刻度」这个数字变了）。",[16,1487,1488,1489,1643],{},"两者相乘：",[394,1490,1492,1520],{"className":1491,"translate":398},[397],[394,1493,1495],{"className":1494},[402],[404,1496,1497],{"xmlns":406},[408,1498,1499,1517],{},[411,1500,1501,1503,1506,1513,1515],{},[806,1502,1428],{},[472,1504,1505],{},"×",[1507,1508,1509,1511],"mfrac",{},[806,1510,970],{},[806,1512,1428],{},[472,1514,1124],{},[806,1516,970],{},[418,1518,1519],{"encoding":420},"100 \\times \\frac{1}{100} = 1",[394,1521,1523,1542,1634],{"className":1522,"ariaHidden":425},[424],[394,1524,1526,1530,1533,1536,1539],{"className":1525},[429],[394,1527],{"className":1528,"style":1529},[433],"height:0.7278em;vertical-align:-0.0833em;",[394,1531,1428],{"className":1532},[438],[394,1534],{"className":1535,"style":440},[1203],[394,1537,1505],{"className":1538},[528],[394,1540],{"className":1541,"style":440},[1203],[394,1543,1545,1549,1625,1628,1631],{"className":1544},[429],[394,1546],{"className":1547,"style":1548},[433],"height:1.1901em;vertical-align:-0.345em;",[394,1550,1552,1556,1622],{"className":1551},[438],[394,1553],{"className":1554},[728,1555],"nulldelimiter",[394,1557,1559],{"className":1558},[1507],[394,1560,1562,1613],{"className":1561},[501,860],[394,1563,1565,1610],{"className":1564},[505],[394,1566,1569,1584,1595],{"className":1567,"style":1568},[509],"height:0.8451em;",[394,1570,1572,1575],{"style":1571},"top:-2.655em;",[394,1573],{"className":1574,"style":874},[516],[394,1576,1578],{"className":1577},[521,522,523,524],[394,1579,1581],{"className":1580},[438,524],[394,1582,1428],{"className":1583},[438,524],[394,1585,1587,1590],{"style":1586},"top:-3.23em;",[394,1588],{"className":1589,"style":874},[516],[394,1591],{"className":1592,"style":1594},[1593],"frac-line","border-bottom-width:0.04em;",[394,1596,1598,1601],{"style":1597},"top:-3.394em;",[394,1599],{"className":1600,"style":874},[516],[394,1602,1604],{"className":1603},[521,522,523,524],[394,1605,1607],{"className":1606},[438,524],[394,1608,970],{"className":1609},[438,524],[394,1611,897],{"className":1612},[896],[394,1614,1616],{"className":1615},[505],[394,1617,1620],{"className":1618,"style":1619},[509],"height:0.345em;",[394,1621],{},[394,1623],{"className":1624},[735,1555],[394,1626],{"className":1627,"style":1204},[1203],[394,1629,1124],{"className":1630},[1208],[394,1632],{"className":1633,"style":1204},[1203],[394,1635,1637,1640],{"className":1636},[429],[394,1638],{"className":1639,"style":1398},[433],[394,1641,970],{"className":1642},[438],"，配对的结果——那个实数——保持不变。",[16,1645,1646],{},"这正是「逆变」（向量，其分量与基的变换方向相反）与「协变」（对偶向量，其分量与基的变换方向相同）这两个物理学名词的来历。对偶空间存在的全部理由，就是要保证最终算出的那个数——无论是长度、功、能量还是概率——是一个客观事实，不依赖于你选择米还是厘米，也不依赖于你采用哪套坐标系。",[16,1648,1649],{},"这也是全篇最重要的一句话：向量与对偶向量构成互补的一对；它们各自的分量随坐标选择而变化，但二者配对得到的标量永远不变。张量理论的每一项技术细节，归根结底都在为守护这个不变性服务。",[11,1651,1652],{"id":1652},"张量的严格定义",[16,1654,1655],{},"向量和对偶向量就位之后，张量的定义便自然浮现：",[32,1657,1658],{},[16,1659,1660,1715,1716,1746,1747,1776,1777,1805],{},[394,1661,1663,1687],{"className":1662,"translate":398},[397],[394,1664,1666],{"className":1665},[402],[404,1667,1668],{"xmlns":406},[408,1669,1670,1684],{},[411,1671,1672,1674,1676,1679,1682],{},[472,1673,703],{"stretchy":702},[414,1675,16],{},[472,1677,1678],{"separator":425},",",[414,1680,1681],{},"q",[472,1683,708],{"stretchy":702},[418,1685,1686],{"encoding":420},"(p,q)",[394,1688,1690],{"className":1689,"ariaHidden":425},[424],[394,1691,1693,1696,1699,1702,1706,1709,1712],{"className":1692},[429],[394,1694],{"className":1695,"style":721},[433],[394,1697,703],{"className":1698},[728],[394,1700,16],{"className":1701},[438,439],[394,1703,1678],{"className":1704},[1705],"mpunct",[394,1707],{"className":1708,"style":1316},[1203],[394,1710,1681],{"className":1711,"style":649},[438,439],[394,1713,708],{"className":1714},[735]," 型张量 ",[394,1717,1719,1733],{"className":1718,"translate":398},[397],[394,1720,1722],{"className":1721},[402],[404,1723,1724],{"xmlns":406},[408,1725,1726,1731],{},[411,1727,1728],{},[414,1729,1730],{},"T",[418,1732,1730],{"encoding":420},[394,1734,1736],{"className":1735,"ariaHidden":425},[424],[394,1737,1739,1742],{"className":1738},[429],[394,1740],{"className":1741,"style":434},[433],[394,1743,1730],{"className":1744,"style":1745},[438,439],"margin-right:0.1389em;"," 是一个多重线性映射：它同时接受 ",[394,1748,1750,1763],{"className":1749,"translate":398},[397],[394,1751,1753],{"className":1752},[402],[404,1754,1755],{"xmlns":406},[408,1756,1757,1761],{},[411,1758,1759],{},[414,1760,16],{},[418,1762,16],{"encoding":420},[394,1764,1766],{"className":1765,"ariaHidden":425},[424],[394,1767,1769,1773],{"className":1768},[429],[394,1770],{"className":1771,"style":1772},[433],"height:0.625em;vertical-align:-0.1944em;",[394,1774,16],{"className":1775},[438,439]," 个对偶向量和 ",[394,1778,1780,1793],{"className":1779,"translate":398},[397],[394,1781,1783],{"className":1782},[402],[404,1784,1785],{"xmlns":406},[408,1786,1787,1791],{},[411,1788,1789],{},[414,1790,1681],{},[418,1792,1681],{"encoding":420},[394,1794,1796],{"className":1795,"ariaHidden":425},[424],[394,1797,1799,1802],{"className":1798},[429],[394,1800],{"className":1801,"style":1772},[433],[394,1803,1681],{"className":1804,"style":649},[438,439]," 个向量作为参数，输出一个实数。",[16,1807,1808],{},"用符号表示：",[394,1810,1812],{"className":1811,"translate":398},[1100],[394,1813,1815,1892],{"className":1814,"translate":398},[397],[394,1816,1818],{"className":1817},[402],[404,1819,1820],{"xmlns":406,"display":1109},[408,1821,1822,1889],{},[411,1823,1824,1826,1829,1860,1862,1882,1885],{},[414,1825,1730],{},[472,1827,1828],{},":",[1830,1831,1832,1858],"munder",{},[1830,1833,1834,1855],{},[411,1835,1836,1842,1844,1847,1849],{},[467,1837,1838,1840],{},[414,1839,416],{},[472,1841,474],{},[472,1843,1505],{},[472,1845,1846],{},"⋯",[472,1848,1505],{},[467,1850,1851,1853],{},[414,1852,416],{},[472,1854,474],{},[472,1856,1857],{"stretchy":425},"⏟",[414,1859,16],{},[472,1861,1505],{},[1830,1863,1864,1880],{},[1830,1865,1866,1878],{},[411,1867,1868,1870,1872,1874,1876],{},[414,1869,416],{},[472,1871,1505],{},[472,1873,1846],{},[472,1875,1505],{},[414,1877,416],{},[472,1879,1857],{"stretchy":425},[414,1881,1681],{},[472,1883,1884],{},"⟶",[414,1886,1888],{"mathvariant":1887},"double-struck","R",[418,1890,1891],{"encoding":420},"T: \\underbrace{V^* \\times \\cdots \\times V^*}_{p} \\times \\underbrace{V \\times \\cdots \\times V}_{q} \\longrightarrow \\mathbb{R}",[394,1893,1895,1913,2137,2283],{"className":1894,"ariaHidden":425},[424],[394,1896,1898,1901,1904,1907,1910],{"className":1897},[429],[394,1899],{"className":1900,"style":434},[433],[394,1902,1730],{"className":1903,"style":1745},[438,439],[394,1905],{"className":1906,"style":1204},[1203],[394,1908,1828],{"className":1909},[1208],[394,1911],{"className":1912,"style":1204},[1203],[394,1914,1916,1920,2128,2131,2134],{"className":1915},[429],[394,1917],{"className":1918,"style":1919},[433],"height:2.1075em;vertical-align:-1.3688em;",[394,1921,1923],{"className":1922},[837,1830],[394,1924,1926,2119],{"className":1925},[501,860],[394,1927,1929,2116],{"className":1928},[505],[394,1930,1933,1948],{"className":1931,"style":1932},[509],"height:0.7387em;",[394,1934,1936,1939],{"style":1935},"top:-1.7673em;",[394,1937],{"className":1938,"style":874},[516],[394,1940,1942],{"className":1941},[521,522,523,524],[394,1943,1945],{"className":1944},[438,524],[394,1946,16],{"className":1947},[438,439,524],[394,1949,1951,1954],{"style":1950},"top:-3em;",[394,1952],{"className":1953,"style":874},[516],[394,1955,1957],{"className":1956},[837,1830],[394,1958,1960,2107],{"className":1959},[501,860],[394,1961,1963,2104],{"className":1962},[505],[394,1964,1966,2016],{"className":1965,"style":1932},[509],[394,1967,1971,1974],{"className":1968,"style":1970},[1969],"svg-align","top:-2.2687em;",[394,1972],{"className":1973,"style":874},[516],[394,1975,1979,1996,2006],{"className":1976,"style":1978},[1977],"stretchy","height:0.548em;min-width:1.6em;",[394,1980,1984],{"className":1981,"style":1983},[1982],"brace-left","height:0.548em;",[1985,1986,1992],"svg",{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":1991},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 0 400000 548","xMinYMin slice",[1993,1994],"path",{"d":1995},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[394,1997,2000],{"className":1998,"style":1983},[1999],"brace-center",[1985,2001,2003],{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":2002},"xMidYMin slice",[1993,2004],{"d":2005},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[394,2007,2010],{"className":2008,"style":1983},[2009],"brace-right",[1985,2011,2013],{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":2012},"xMaxYMin slice",[1993,2014],{"d":2015},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[394,2017,2018,2021],{"style":1950},[394,2019],{"className":2020,"style":874},[516],[394,2022,2024,2054,2057,2060,2063,2066,2069,2072,2075],{"className":2023},[438],[394,2025,2027,2030],{"className":2026},[438],[394,2028,416],{"className":2029,"style":440},[438,439],[394,2031,2033],{"className":2032},[497],[394,2034,2036],{"className":2035},[501],[394,2037,2039],{"className":2038},[505],[394,2040,2042],{"className":2041,"style":1932},[509],[394,2043,2045,2048],{"style":2044},"top:-3.113em;margin-right:0.05em;",[394,2046],{"className":2047,"style":517},[516],[394,2049,2051],{"className":2050},[521,522,523,524],[394,2052,474],{"className":2053},[528,524],[394,2055],{"className":2056,"style":440},[1203],[394,2058,1505],{"className":2059},[528],[394,2061],{"className":2062,"style":440},[1203],[394,2064,1846],{"className":2065},[837],[394,2067],{"className":2068,"style":440},[1203],[394,2070,1505],{"className":2071},[528],[394,2073],{"className":2074,"style":440},[1203],[394,2076,2078,2081],{"className":2077},[438],[394,2079,416],{"className":2080,"style":440},[438,439],[394,2082,2084],{"className":2083},[497],[394,2085,2087],{"className":2086},[501],[394,2088,2090],{"className":2089},[505],[394,2091,2093],{"className":2092,"style":1932},[509],[394,2094,2095,2098],{"style":2044},[394,2096],{"className":2097,"style":517},[516],[394,2099,2101],{"className":2100},[521,522,523,524],[394,2102,474],{"className":2103},[528,524],[394,2105,897],{"className":2106},[896],[394,2108,2110],{"className":2109},[505],[394,2111,2114],{"className":2112,"style":2113},[509],"height:0.7313em;",[394,2115],{},[394,2117,897],{"className":2118},[896],[394,2120,2122],{"className":2121},[505],[394,2123,2126],{"className":2124,"style":2125},[509],"height:1.3688em;",[394,2127],{},[394,2129],{"className":2130,"style":440},[1203],[394,2132,1505],{"className":2133},[528],[394,2135],{"className":2136,"style":440},[1203],[394,2138,2140,2144,2274,2277,2280],{"className":2139},[429],[394,2141],{"className":2142,"style":2143},[433],"height:2.0522em;vertical-align:-1.3688em;",[394,2145,2147],{"className":2146},[837,1830],[394,2148,2150,2266],{"className":2149},[501,860],[394,2151,2153,2263],{"className":2152},[505],[394,2154,2156,2170],{"className":2155,"style":434},[509],[394,2157,2158,2161],{"style":1935},[394,2159],{"className":2160,"style":874},[516],[394,2162,2164],{"className":2163},[521,522,523,524],[394,2165,2167],{"className":2166},[438,524],[394,2168,1681],{"className":2169,"style":649},[438,439,524],[394,2171,2172,2175],{"style":1950},[394,2173],{"className":2174,"style":874},[516],[394,2176,2178],{"className":2177},[837,1830],[394,2179,2181,2255],{"className":2180},[501,860],[394,2182,2184,2252],{"className":2183},[505],[394,2185,2187,2217],{"className":2186,"style":434},[509],[394,2188,2190,2193],{"className":2189,"style":1970},[1969],[394,2191],{"className":2192,"style":874},[516],[394,2194,2196,2203,2210],{"className":2195,"style":1978},[1977],[394,2197,2199],{"className":2198,"style":1983},[1982],[1985,2200,2201],{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":1991},[1993,2202],{"d":1995},[394,2204,2206],{"className":2205,"style":1983},[1999],[1985,2207,2208],{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":2002},[1993,2209],{"d":2005},[394,2211,2213],{"className":2212,"style":1983},[2009],[1985,2214,2215],{"xmlns":1987,"width":1988,"height":1989,"viewBox":1990,"preserveAspectRatio":2012},[1993,2216],{"d":2015},[394,2218,2219,2222],{"style":1950},[394,2220],{"className":2221,"style":874},[516],[394,2223,2225,2228,2231,2234,2237,2240,2243,2246,2249],{"className":2224},[438],[394,2226,416],{"className":2227,"style":440},[438,439],[394,2229],{"className":2230,"style":440},[1203],[394,2232,1505],{"className":2233},[528],[394,2235],{"className":2236,"style":440},[1203],[394,2238,1846],{"className":2239},[837],[394,2241],{"className":2242,"style":440},[1203],[394,2244,1505],{"className":2245},[528],[394,2247],{"className":2248,"style":440},[1203],[394,2250,416],{"className":2251,"style":440},[438,439],[394,2253,897],{"className":2254},[896],[394,2256,2258],{"className":2257},[505],[394,2259,2261],{"className":2260,"style":2113},[509],[394,2262],{},[394,2264,897],{"className":2265},[896],[394,2267,2269],{"className":2268},[505],[394,2270,2272],{"className":2271,"style":2125},[509],[394,2273],{},[394,2275],{"className":2276,"style":1204},[1203],[394,2278,1884],{"className":2279},[1208],[394,2281],{"className":2282,"style":1204},[1203],[394,2284,2286,2290],{"className":2285},[429],[394,2287],{"className":2288,"style":2289},[433],"height:0.6889em;",[394,2291,1888],{"className":2292},[438,2293],"mathbb",[16,2295,2296],{},"两个关键词，一个都不能少：",[97,2298,2299,2302],{},[100,2300,2301],{},"多重线性：对每个槽位分别线性（加法保持、对标量乘法齐次）。这保证张量运算可以像矩阵运算一样，完全展开为分量的加权和。",[100,2303,2304],{},"标量输出：无论有多少个槽位、输入的是什么类型，最终出来的都是一个不随坐标变化而变化的绝对数字。",[16,2306,2307],{},"由此直接得到张量阶数的公式：",[394,2309,2311],{"className":2310,"translate":398},[1100],[394,2312,2314,2339],{"className":2313,"translate":398},[397],[394,2315,2317],{"className":2316},[402],[404,2318,2319],{"xmlns":406,"display":1109},[408,2320,2321,2336],{},[411,2322,2323,2327,2329,2331,2334],{},[2324,2325,2326],"mtext",{},"阶数",[472,2328,1124],{},[414,2330,16],{},[472,2332,2333],{},"+",[414,2335,1681],{},[418,2337,2338],{"encoding":420},"\\text{阶数} = p + q",[394,2340,2342,2365,2384],{"className":2341,"ariaHidden":425},[424],[394,2343,2345,2348,2356,2359,2362],{"className":2344},[429],[394,2346],{"className":2347,"style":434},[433],[394,2349,2352],{"className":2350},[438,2351],"text",[394,2353,2326],{"className":2354},[438,2355],"cjk_fallback",[394,2357],{"className":2358,"style":1204},[1203],[394,2360,1124],{"className":2361},[1208],[394,2363],{"className":2364,"style":1204},[1203],[394,2366,2368,2372,2375,2378,2381],{"className":2367},[429],[394,2369],{"className":2370,"style":2371},[433],"height:0.7778em;vertical-align:-0.1944em;",[394,2373,16],{"className":2374},[438,439],[394,2376],{"className":2377,"style":440},[1203],[394,2379,2333],{"className":2380},[528],[394,2382],{"className":2383,"style":440},[1203],[394,2385,2387,2390],{"className":2386},[429],[394,2388],{"className":2389,"style":1772},[433],[394,2391,1681],{"className":2392,"style":649},[438,439],[16,2394,2395],{},"这也是为什么单说「阶数」信息量不够——它只告诉你张量有几个槽位，却不告诉你哪些槽位吃向量、哪些槽位吃对偶向量。这种区分绝非咬文嚼字；下一章将用一个具体例子说明，为什么它是生死攸关的问题。",[11,2397,2398],{"id":2398},"槽位哲学",[16,2400,2401],{},"物理教科书常常这样描述二阶张量：",[32,2403,2404],{},[16,2405,2406,2407,2437,2438,2468,2469,2500],{},"「应力张量 ",[394,2408,2410,2425],{"className":2409,"translate":398},[397],[394,2411,2413],{"className":2412},[402],[404,2414,2415],{"xmlns":406},[408,2416,2417,2422],{},[411,2418,2419],{},[414,2420,2421],{},"σ",[418,2423,2424],{"encoding":420},"\\sigma",[394,2426,2428],{"className":2427,"ariaHidden":425},[424],[394,2429,2431,2434],{"className":2430},[429],[394,2432],{"className":2433,"style":645},[433],[394,2435,2421],{"className":2436,"style":649},[438,439]," 作用在一个法向量 ",[394,2439,2441,2456],{"className":2440,"translate":398},[397],[394,2442,2444],{"className":2443},[402],[404,2445,2446],{"xmlns":406},[408,2447,2448,2453],{},[411,2449,2450],{},[414,2451,2452],{"mathvariant":665},"n",[418,2454,2455],{"encoding":420},"\\mathbf{n}",[394,2457,2459],{"className":2458,"ariaHidden":425},[424],[394,2460,2462,2465],{"className":2461},[429],[394,2463],{"className":2464,"style":679},[433],[394,2466,2452],{"className":2467},[438,683]," 上，产生一个应力向量 ",[394,2470,2472,2487],{"className":2471,"translate":398},[397],[394,2473,2475],{"className":2474},[402],[404,2476,2477],{"xmlns":406},[408,2478,2479,2484],{},[411,2480,2481],{},[414,2482,2483],{"mathvariant":665},"t",[418,2485,2486],{"encoding":420},"\\mathbf{t}",[394,2488,2490],{"className":2489,"ariaHidden":425},[424],[394,2491,2493,2497],{"className":2492},[429],[394,2494],{"className":2495,"style":2496},[433],"height:0.6349em;",[394,2498,2483],{"className":2499},[438,683],"。」",[16,2502,2503],{},"这句话听起来像是「输入一个向量，输出一个向量」。而第 3 章的定义明明说，张量是「输入几个向量和对偶向量、输出一个标量」的东西。两种说法似乎完全不可调和——而这正是绝大多数读者卡壳的地方。",[16,2505,2506],{},"解开这个矛盾的钥匙是：多重线性映射并不要求你一次填满所有槽位。",[16,2508,2509,2510,529,2538,2590,2591,2619],{},"应力张量 ",[394,2511,2513,2526],{"className":2512,"translate":398},[397],[394,2514,2516],{"className":2515},[402],[404,2517,2518],{"xmlns":406},[408,2519,2520,2524],{},[411,2521,2522],{},[414,2523,2421],{},[418,2525,2424],{"encoding":420},[394,2527,2529],{"className":2528,"ariaHidden":425},[424],[394,2530,2532,2535],{"className":2531},[429],[394,2533],{"className":2534,"style":645},[433],[394,2536,2421],{"className":2537,"style":649},[438,439],[394,2539,2541,2563],{"className":2540,"translate":398},[397],[394,2542,2544],{"className":2543},[402],[404,2545,2546],{"xmlns":406},[408,2547,2548,2560],{},[411,2549,2550,2552,2554,2556,2558],{},[472,2551,703],{"stretchy":702},[806,2553,970],{},[472,2555,1678],{"separator":425},[806,2557,970],{},[472,2559,708],{"stretchy":702},[418,2561,2562],{"encoding":420},"(1,1)",[394,2564,2566],{"className":2565,"ariaHidden":425},[424],[394,2567,2569,2572,2575,2578,2581,2584,2587],{"className":2568},[429],[394,2570],{"className":2571,"style":721},[433],[394,2573,703],{"className":2574},[728],[394,2576,970],{"className":2577},[438],[394,2579,1678],{"className":2580},[1705],[394,2582],{"className":2583,"style":1316},[1203],[394,2585,970],{"className":2586},[438],[394,2588,708],{"className":2589},[735]," 型张量，有两个槽位：一个对偶向量槽位和一个向量槽位。当你只把法向量 ",[394,2592,2594,2607],{"className":2593,"translate":398},[397],[394,2595,2597],{"className":2596},[402],[404,2598,2599],{"xmlns":406},[408,2600,2601,2605],{},[411,2602,2603],{},[414,2604,2452],{"mathvariant":665},[418,2606,2455],{"encoding":420},[394,2608,2610],{"className":2609,"ariaHidden":425},[424],[394,2611,2613,2616],{"className":2612},[429],[394,2614],{"className":2615,"style":679},[433],[394,2617,2452],{"className":2618},[438,683]," 塞进它的向量槽位时：",[97,2621,2622,2625],{},[100,2623,2624],{},"向量槽位已被填满；",[100,2626,2627],{},"对偶向量槽位仍然空着。",[16,2629,2630,2631,2689,2690,1444],{},"此时 ",[394,2632,2634,2659],{"className":2633,"translate":398},[397],[394,2635,2637],{"className":2636},[402],[404,2638,2639],{"xmlns":406},[408,2640,2641,2656],{},[411,2642,2643,2645,2647,2650,2652,2654],{},[414,2644,2421],{},[472,2646,703],{"stretchy":702},[472,2648,2649],{},"⋅",[472,2651,1678],{"separator":425},[414,2653,2452],{"mathvariant":665},[472,2655,708],{"stretchy":702},[418,2657,2658],{"encoding":420},"\\sigma(\\cdot, \\mathbf{n})",[394,2660,2662],{"className":2661,"ariaHidden":425},[424],[394,2663,2665,2668,2671,2674,2677,2680,2683,2686],{"className":2664},[429],[394,2666],{"className":2667,"style":721},[433],[394,2669,2421],{"className":2670,"style":649},[438,439],[394,2672,703],{"className":2673},[728],[394,2675,2649],{"className":2676},[438],[394,2678,1678],{"className":2679},[1705],[394,2681],{"className":2682,"style":1316},[1203],[394,2684,2452],{"className":2685},[438,683],[394,2687,708],{"className":2688},[735]," 变成了什么？它变成了一个仍在等待对偶向量输入的中途产物——而「一台等待对偶向量、随后输出标量的机器」，恰好是定义向量的另一种严格方式（通过自然同构 ",[394,2691,2693,2720],{"className":2692,"translate":398},[397],[394,2694,2696],{"className":2695},[402],[404,2697,2698],{"xmlns":406},[408,2699,2700,2717],{},[411,2701,2702,2704,2707],{},[414,2703,416],{},[472,2705,2706],{},"≅",[467,2708,2709,2711],{},[414,2710,416],{},[411,2712,2713,2715],{},[472,2714,474],{},[472,2716,474],{},[418,2718,2719],{"encoding":420},"V \\cong V^{**}",[394,2721,2723,2741],{"className":2722,"ariaHidden":425},[424],[394,2724,2726,2729,2732,2735,2738],{"className":2725},[429],[394,2727],{"className":2728,"style":434},[433],[394,2730,416],{"className":2731,"style":440},[438,439],[394,2733],{"className":2734,"style":1204},[1203],[394,2736,2706],{"className":2737},[1208],[394,2739],{"className":2740,"style":1204},[1203],[394,2742,2744,2747],{"className":2743},[429],[394,2745],{"className":2746,"style":487},[433],[394,2748,2750,2753],{"className":2749},[438],[394,2751,416],{"className":2752,"style":440},[438,439],[394,2754,2756],{"className":2755},[497],[394,2757,2759],{"className":2758},[501],[394,2760,2762],{"className":2761},[505],[394,2763,2765],{"className":2764,"style":487},[509],[394,2766,2767,2770],{"style":512},[394,2768],{"className":2769,"style":517},[516],[394,2771,2773],{"className":2772},[521,522,523,524],[394,2774,2776],{"className":2775},[438,524],[394,2777,2779],{"className":2778},[438,524],"∗∗",[97,2781,2782,2843],{},[100,2783,2784,2785,2813,2814,2842],{},"只填一个槽位：",[394,2786,2788,2801],{"className":2787,"translate":398},[397],[394,2789,2791],{"className":2790},[402],[404,2792,2793],{"xmlns":406},[408,2794,2795,2799],{},[411,2796,2797],{},[414,2798,2421],{},[418,2800,2424],{"encoding":420},[394,2802,2804],{"className":2803,"ariaHidden":425},[424],[394,2805,2807,2810],{"className":2806},[429],[394,2808],{"className":2809,"style":645},[433],[394,2811,2421],{"className":2812,"style":649},[438,439]," 退化成一个「待定」向量，这正是教科书里说的「输出应力向量 ",[394,2815,2817,2830],{"className":2816,"translate":398},[397],[394,2818,2820],{"className":2819},[402],[404,2821,2822],{"xmlns":406},[408,2823,2824,2828],{},[411,2825,2826],{},[414,2827,2483],{"mathvariant":665},[418,2829,2486],{"encoding":420},[394,2831,2833],{"className":2832,"ariaHidden":425},[424],[394,2834,2836,2839],{"className":2835},[429],[394,2837],{"className":2838,"style":2496},[433],[394,2840,2483],{"className":2841},[438,683],"」；",[100,2844,2845],{},"两个槽位都填满（法向量 + 某个方向上的对偶向量\u002F标尺）：张量终于输出一个标量，即该方向上应力分量的大小。",[16,2847,2848],{},"因此，「输入一个向量、输出一个向量」与「输入两个参数、输出一个标量」并不矛盾；前者只是后者的中间状态。物理教科书为了照顾直觉，通常只展示「喂了一半」的结果；数学定义则严谨地描述所有槽位填满后的最终行为。理解了这一层，关于张量「到底是映射还是运算」的争论几乎会自行消解。",[16,2850,2851,2852,2880,2881,2909],{},"那么，为什么必须区分 ",[394,2853,2855,2868],{"className":2854,"translate":398},[397],[394,2856,2858],{"className":2857},[402],[404,2859,2860],{"xmlns":406},[408,2861,2862,2866],{},[411,2863,2864],{},[414,2865,16],{},[418,2867,16],{"encoding":420},[394,2869,2871],{"className":2870,"ariaHidden":425},[424],[394,2872,2874,2877],{"className":2873},[429],[394,2875],{"className":2876,"style":1772},[433],[394,2878,16],{"className":2879},[438,439]," 和 ",[394,2882,2884,2897],{"className":2883,"translate":398},[397],[394,2885,2887],{"className":2886},[402],[404,2888,2889],{"xmlns":406},[408,2890,2891,2895],{},[411,2892,2893],{},[414,2894,1681],{},[418,2896,1681],{"encoding":420},[394,2898,2900],{"className":2899,"ariaHidden":425},[424],[394,2901,2903,2906],{"className":2902},[429],[394,2904],{"className":2905,"style":1772},[433],[394,2907,1681],{"className":2908,"style":649},[438,439],"，而不是只说阶数？",[16,2911,2912,2963,2964,3016],{},[394,2913,2915,2936],{"className":2914,"translate":398},[397],[394,2916,2918],{"className":2917},[402],[404,2919,2920],{"xmlns":406},[408,2921,2922,2934],{},[411,2923,2924,2926,2928,2930,2932],{},[472,2925,703],{"stretchy":702},[806,2927,970],{},[472,2929,1678],{"separator":425},[806,2931,970],{},[472,2933,708],{"stretchy":702},[418,2935,2562],{"encoding":420},[394,2937,2939],{"className":2938,"ariaHidden":425},[424],[394,2940,2942,2945,2948,2951,2954,2957,2960],{"className":2941},[429],[394,2943],{"className":2944,"style":721},[433],[394,2946,703],{"className":2947},[728],[394,2949,970],{"className":2950},[438],[394,2952,1678],{"className":2953},[1705],[394,2955],{"className":2956,"style":1316},[1203],[394,2958,970],{"className":2959},[438],[394,2961,708],{"className":2962},[735]," 型张量与 ",[394,2965,2967,2989],{"className":2966,"translate":398},[397],[394,2968,2970],{"className":2969},[402],[404,2971,2972],{"xmlns":406},[408,2973,2974,2986],{},[411,2975,2976,2978,2980,2982,2984],{},[472,2977,703],{"stretchy":702},[806,2979,804],{},[472,2981,1678],{"separator":425},[806,2983,808],{},[472,2985,708],{"stretchy":702},[418,2987,2988],{"encoding":420},"(0,2)",[394,2990,2992],{"className":2991,"ariaHidden":425},[424],[394,2993,2995,2998,3001,3004,3007,3010,3013],{"className":2994},[429],[394,2996],{"className":2997,"style":721},[433],[394,2999,703],{"className":3000},[728],[394,3002,804],{"className":3003},[438],[394,3005,1678],{"className":3006},[1705],[394,3008],{"className":3009,"style":1316},[1203],[394,3011,808],{"className":3012},[438],[394,3014,708],{"className":3015},[735]," 型张量，写成数组时都是二维矩阵——视觉上无法区分。但它们在坐标变换下的行为截然不同：",[97,3018,3019,3073],{},[100,3020,3021,3072],{},[394,3022,3024,3045],{"className":3023,"translate":398},[397],[394,3025,3027],{"className":3026},[402],[404,3028,3029],{"xmlns":406},[408,3030,3031,3043],{},[411,3032,3033,3035,3037,3039,3041],{},[472,3034,703],{"stretchy":702},[806,3036,970],{},[472,3038,1678],{"separator":425},[806,3040,970],{},[472,3042,708],{"stretchy":702},[418,3044,2562],{"encoding":420},[394,3046,3048],{"className":3047,"ariaHidden":425},[424],[394,3049,3051,3054,3057,3060,3063,3066,3069],{"className":3050},[429],[394,3052],{"className":3053,"style":721},[433],[394,3055,703],{"className":3056},[728],[394,3058,970],{"className":3059},[438],[394,3061,1678],{"className":3062},[1705],[394,3064],{"className":3065,"style":1316},[1203],[394,3067,970],{"className":3068},[438],[394,3070,708],{"className":3071},[735]," 型（例如线性变换、惯性张量）：一个上标一个下标；变换时一个分量「逆变」、一个分量「协变」，两者相互抵消。",[100,3074,3075,3126],{},[394,3076,3078,3099],{"className":3077,"translate":398},[397],[394,3079,3081],{"className":3080},[402],[404,3082,3083],{"xmlns":406},[408,3084,3085,3097],{},[411,3086,3087,3089,3091,3093,3095],{},[472,3088,703],{"stretchy":702},[806,3090,804],{},[472,3092,1678],{"separator":425},[806,3094,808],{},[472,3096,708],{"stretchy":702},[418,3098,2988],{"encoding":420},[394,3100,3102],{"className":3101,"ariaHidden":425},[424],[394,3103,3105,3108,3111,3114,3117,3120,3123],{"className":3104},[429],[394,3106],{"className":3107,"style":721},[433],[394,3109,703],{"className":3110},[728],[394,3112,804],{"className":3113},[438],[394,3115,1678],{"className":3116},[1705],[394,3118],{"className":3119,"style":1316},[1203],[394,3121,808],{"className":3122},[438],[394,3124,708],{"className":3125},[735]," 型（例如度量张量）：两个下标；变换时两个分量都「协变」，向同一方向累积。",[16,3128,3129,3130,3182,3183,3234],{},"如果只说「阶数」而忽略 ",[394,3131,3133,3155],{"className":3132,"translate":398},[397],[394,3134,3136],{"className":3135},[402],[404,3137,3138],{"xmlns":406},[408,3139,3140,3152],{},[411,3141,3142,3144,3146,3148,3150],{},[472,3143,703],{"stretchy":702},[414,3145,16],{},[472,3147,1678],{"separator":425},[414,3149,1681],{},[472,3151,708],{"stretchy":702},[418,3153,3154],{"encoding":420},"(p, q)",[394,3156,3158],{"className":3157,"ariaHidden":425},[424],[394,3159,3161,3164,3167,3170,3173,3176,3179],{"className":3160},[429],[394,3162],{"className":3163,"style":721},[433],[394,3165,703],{"className":3166},[728],[394,3168,16],{"className":3169},[438,439],[394,3171,1678],{"className":3172},[1705],[394,3174],{"className":3175,"style":1316},[1203],[394,3177,1681],{"className":3178,"style":649},[438,439],[394,3180,708],{"className":3181},[735]," 之分，在计算散度、缩并等运算时必然误套公式——而在广义相对论里，这种错误足以让「能量守恒」这样的物理定律在计算层面崩塌。这就是数学家坚持用 ",[394,3184,3186,3207],{"className":3185,"translate":398},[397],[394,3187,3189],{"className":3188},[402],[404,3190,3191],{"xmlns":406},[408,3192,3193,3205],{},[411,3194,3195,3197,3199,3201,3203],{},[472,3196,703],{"stretchy":702},[414,3198,16],{},[472,3200,1678],{"separator":425},[414,3202,1681],{},[472,3204,708],{"stretchy":702},[418,3206,1686],{"encoding":420},[394,3208,3210],{"className":3209,"ariaHidden":425},[424],[394,3211,3213,3216,3219,3222,3225,3228,3231],{"className":3212},[429],[394,3214],{"className":3215,"style":721},[433],[394,3217,703],{"className":3218},[728],[394,3220,16],{"className":3221},[438,439],[394,3223,1678],{"className":3224},[1705],[394,3226],{"className":3227,"style":1316},[1203],[394,3229,1681],{"className":3230,"style":649},[438,439],[394,3232,708],{"className":3233},[735]," 型而非笼统的「阶数」来标注张量的原因。",[11,3236,3237],{"id":3237},"度量张量",[16,3239,3240],{},"至此还有一个问题悬而未决：向量和对偶向量居于两个显然不同的空间，物理学家却经常把二者混用（比如说「把一个向量的指标降下来」）。这到底是怎么回事？",[16,3242,3243,3244,3331],{},"在一个配有度量张量 ",[394,3245,3247,3273],{"className":3246,"translate":398},[397],[394,3248,3250],{"className":3249},[402],[404,3251,3252],{"xmlns":406},[408,3253,3254,3270],{},[411,3255,3256],{},[3257,3258,3259,3262],"msub",{},[414,3260,3261],{},"g",[411,3263,3264,3267],{},[414,3265,3266],{},"i",[414,3268,3269],{},"j",[418,3271,3272],{"encoding":420},"g_{ij}",[394,3274,3276],{"className":3275,"ariaHidden":425},[424],[394,3277,3279,3283],{"className":3278},[429],[394,3280],{"className":3281,"style":3282},[433],"height:0.7167em;vertical-align:-0.2861em;",[394,3284,3286,3289],{"className":3285},[438],[394,3287,3261],{"className":3288,"style":649},[438,439],[394,3290,3292],{"className":3291},[497],[394,3293,3295,3322],{"className":3294},[501,860],[394,3296,3298,3319],{"className":3297},[505],[394,3299,3302],{"className":3300,"style":3301},[509],"height:0.3117em;",[394,3303,3305,3308],{"style":3304},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[394,3306],{"className":3307,"style":517},[516],[394,3309,3311],{"className":3310},[521,522,523,524],[394,3312,3314],{"className":3313},[438,524],[394,3315,3318],{"className":3316,"style":3317},[438,439,524],"margin-right:0.0572em;","ij",[394,3320,897],{"className":3321},[896],[394,3323,3325],{"className":3324},[505],[394,3326,3329],{"className":3327,"style":3328},[509],"height:0.2861em;",[394,3330],{}," 的空间里（几乎所有的物理空间都是如此），存在一台自然的「翻译机」，能把向量翻译成对偶向量，也能反向翻译。这个操作叫作指标升降：",[16,3333,3334],{},"降指标：向量到对偶向量",[394,3336,3338],{"className":3337,"translate":398},[1100],[394,3339,3341,3377],{"className":3340,"translate":398},[397],[394,3342,3344],{"className":3343},[402],[404,3345,3346],{"xmlns":406,"display":1109},[408,3347,3348,3374],{},[411,3349,3350,3356,3358,3368],{},[3257,3351,3352,3354],{},[414,3353,666],{},[414,3355,3266],{},[472,3357,1124],{},[3257,3359,3360,3362],{},[414,3361,3261],{},[411,3363,3364,3366],{},[414,3365,3266],{},[414,3367,3269],{},[467,3369,3370,3372],{},[414,3371,666],{},[414,3373,3269],{},[418,3375,3376],{"encoding":420},"v_i = g_{ij} v^j",[394,3378,3380,3437],{"className":3379,"ariaHidden":425},[424],[394,3381,3383,3387,3428,3431,3434],{"className":3382},[429],[394,3384],{"className":3385,"style":3386},[433],"height:0.5806em;vertical-align:-0.15em;",[394,3388,3390,3393],{"className":3389},[438],[394,3391,666],{"className":3392,"style":649},[438,439],[394,3394,3396],{"className":3395},[497],[394,3397,3399,3419],{"className":3398},[501,860],[394,3400,3402,3416],{"className":3401},[505],[394,3403,3405],{"className":3404,"style":3301},[509],[394,3406,3407,3410],{"style":3304},[394,3408],{"className":3409,"style":517},[516],[394,3411,3413],{"className":3412},[521,522,523,524],[394,3414,3266],{"className":3415},[438,439,524],[394,3417,897],{"className":3418},[896],[394,3420,3422],{"className":3421},[505],[394,3423,3426],{"className":3424,"style":3425},[509],"height:0.15em;",[394,3427],{},[394,3429],{"className":3430,"style":1204},[1203],[394,3432,1124],{"className":3433},[1208],[394,3435],{"className":3436,"style":1204},[1203],[394,3438,3440,3444,3487],{"className":3439},[429],[394,3441],{"className":3442,"style":3443},[433],"height:1.1608em;vertical-align:-0.2861em;",[394,3445,3447,3450],{"className":3446},[438],[394,3448,3261],{"className":3449,"style":649},[438,439],[394,3451,3453],{"className":3452},[497],[394,3454,3456,3479],{"className":3455},[501,860],[394,3457,3459,3476],{"className":3458},[505],[394,3460,3462],{"className":3461,"style":3301},[509],[394,3463,3464,3467],{"style":3304},[394,3465],{"className":3466,"style":517},[516],[394,3468,3470],{"className":3469},[521,522,523,524],[394,3471,3473],{"className":3472},[438,524],[394,3474,3318],{"className":3475,"style":3317},[438,439,524],[394,3477,897],{"className":3478},[896],[394,3480,3482],{"className":3481},[505],[394,3483,3485],{"className":3484,"style":3328},[509],[394,3486],{},[394,3488,3490,3493],{"className":3489},[438],[394,3491,666],{"className":3492,"style":649},[438,439],[394,3494,3496],{"className":3495},[497],[394,3497,3499],{"className":3498},[501],[394,3500,3502],{"className":3501},[505],[394,3503,3506],{"className":3504,"style":3505},[509],"height:0.8747em;",[394,3507,3508,3511],{"style":2044},[394,3509],{"className":3510,"style":517},[516],[394,3512,3514],{"className":3513},[521,522,523,524],[394,3515,3269],{"className":3516,"style":3317},[438,439,524],[16,3518,3519],{},"升指标：对偶向量到向量",[394,3521,3523],{"className":3522,"translate":398},[1100],[394,3524,3526,3562],{"className":3525,"translate":398},[397],[394,3527,3529],{"className":3528},[402],[404,3530,3531],{"xmlns":406,"display":1109},[408,3532,3533,3559],{},[411,3534,3535,3541,3543,3553],{},[467,3536,3537,3539],{},[414,3538,666],{},[414,3540,3266],{},[472,3542,1124],{},[467,3544,3545,3547],{},[414,3546,3261],{},[411,3548,3549,3551],{},[414,3550,3266],{},[414,3552,3269],{},[3257,3554,3555,3557],{},[414,3556,666],{},[414,3558,3269],{},[418,3560,3561],{"encoding":420},"v^i = g^{ij} v_j",[394,3563,3565,3609],{"className":3564,"ariaHidden":425},[424],[394,3566,3568,3571,3600,3603,3606],{"className":3567},[429],[394,3569],{"className":3570,"style":3505},[433],[394,3572,3574,3577],{"className":3573},[438],[394,3575,666],{"className":3576,"style":649},[438,439],[394,3578,3580],{"className":3579},[497],[394,3581,3583],{"className":3582},[501],[394,3584,3586],{"className":3585},[505],[394,3587,3589],{"className":3588,"style":3505},[509],[394,3590,3591,3594],{"style":2044},[394,3592],{"className":3593,"style":517},[516],[394,3595,3597],{"className":3596},[521,522,523,524],[394,3598,3266],{"className":3599},[438,439,524],[394,3601],{"className":3602,"style":1204},[1203],[394,3604,1124],{"className":3605},[1208],[394,3607],{"className":3608,"style":1204},[1203],[394,3610,3612,3615,3647],{"className":3611},[429],[394,3613],{"className":3614,"style":3443},[433],[394,3616,3618,3621],{"className":3617},[438],[394,3619,3261],{"className":3620,"style":649},[438,439],[394,3622,3624],{"className":3623},[497],[394,3625,3627],{"className":3626},[501],[394,3628,3630],{"className":3629},[505],[394,3631,3633],{"className":3632,"style":3505},[509],[394,3634,3635,3638],{"style":2044},[394,3636],{"className":3637,"style":517},[516],[394,3639,3641],{"className":3640},[521,522,523,524],[394,3642,3644],{"className":3643},[438,524],[394,3645,3318],{"className":3646,"style":3317},[438,439,524],[394,3648,3650,3653],{"className":3649},[438],[394,3651,666],{"className":3652,"style":649},[438,439],[394,3654,3656],{"className":3655},[497],[394,3657,3659,3679],{"className":3658},[501,860],[394,3660,3662,3676],{"className":3661},[505],[394,3663,3665],{"className":3664,"style":3301},[509],[394,3666,3667,3670],{"style":3304},[394,3668],{"className":3669,"style":517},[516],[394,3671,3673],{"className":3672},[521,522,523,524],[394,3674,3269],{"className":3675,"style":3317},[438,439,524],[394,3677,897],{"className":3678},[896],[394,3680,3682],{"className":3681},[505],[394,3683,3685],{"className":3684,"style":3328},[509],[394,3686],{},[16,3688,3689,3690,3741],{},"度量张量本身是 ",[394,3691,3693,3714],{"className":3692,"translate":398},[397],[394,3694,3696],{"className":3695},[402],[404,3697,3698],{"xmlns":406},[408,3699,3700,3712],{},[411,3701,3702,3704,3706,3708,3710],{},[472,3703,703],{"stretchy":702},[806,3705,804],{},[472,3707,1678],{"separator":425},[806,3709,808],{},[472,3711,708],{"stretchy":702},[418,3713,2988],{"encoding":420},[394,3715,3717],{"className":3716,"ariaHidden":425},[424],[394,3718,3720,3723,3726,3729,3732,3735,3738],{"className":3719},[429],[394,3721],{"className":3722,"style":721},[433],[394,3724,703],{"className":3725},[728],[394,3727,804],{"className":3728},[438],[394,3730,1678],{"className":3731},[1705],[394,3733],{"className":3734,"style":1316},[1203],[394,3736,808],{"className":3737},[438],[394,3739,708],{"className":3740},[735]," 型张量；它的作用正是「吞进两个向量，输出它们的内积」——而内积正是长度与角度的定义。正因有了度量，我们才能谈论「向量的长度」这个听起来像是向量固有属性、实际上离开对偶配对就无法定义的量。",[16,3743,3744],{},"在欧几里得空间（比如熟悉的笛卡尔坐标系）里，度量恰好是单位矩阵，指标升降「看起来」什么也没做。这也是为什么中学物理里向量与对偶向量的区别被完全掩盖了。一旦坐标系变得复杂（极坐标、球坐标），或者空间本身弯曲（广义相对论的时空），度量就不再是单位矩阵，向量与对偶向量的区别就会以决定性的姿态重新登场。",[11,3746,3747],{"id":3747},"坐标变换法则",[16,3749,3750,3751,3780,3781,382,3846,3897],{},"前面几章都在诉诸直觉；这里给出具体的分量变换法则作为「锚」。设坐标从 ",[394,3752,3754,3768],{"className":3753,"translate":398},[397],[394,3755,3757],{"className":3756},[402],[404,3758,3759],{"xmlns":406},[408,3760,3761,3766],{},[411,3762,3763],{},[414,3764,3765],{},"x",[418,3767,3765],{"encoding":420},[394,3769,3771],{"className":3770,"ariaHidden":425},[424],[394,3772,3774,3777],{"className":3773},[429],[394,3775],{"className":3776,"style":645},[433],[394,3778,3765],{"className":3779},[438,439]," 变换到 ",[394,3782,3784,3804],{"className":3783,"translate":398},[397],[394,3785,3787],{"className":3786},[402],[404,3788,3789],{"xmlns":406},[408,3790,3791,3801],{},[411,3792,3793],{},[467,3794,3795,3797],{},[414,3796,3765],{},[472,3798,3800],{"mathvariant":1424,"lspace":3799,"rspace":3799},"0em","′",[418,3802,3803],{"encoding":420},"x'",[394,3805,3807],{"className":3806,"ariaHidden":425},[424],[394,3808,3810,3814],{"className":3809},[429],[394,3811],{"className":3812,"style":3813},[433],"height:0.7519em;",[394,3815,3817,3820],{"className":3816},[438],[394,3818,3765],{"className":3819},[438,439],[394,3821,3823],{"className":3822},[497],[394,3824,3826],{"className":3825},[501],[394,3827,3829],{"className":3828},[505],[394,3830,3832],{"className":3831,"style":3813},[509],[394,3833,3834,3837],{"style":512},[394,3835],{"className":3836,"style":517},[516],[394,3838,3840],{"className":3839},[521,522,523,524],[394,3841,3843],{"className":3842},[438,524],[394,3844,3800],{"className":3845},[438,524],[394,3847,3849,3870],{"className":3848,"translate":398},[397],[394,3850,3852],{"className":3851},[402],[404,3853,3854],{"xmlns":406},[408,3855,3856,3868],{},[411,3857,3858,3860,3862,3864,3866],{},[472,3859,703],{"stretchy":702},[414,3861,16],{},[472,3863,1678],{"separator":425},[414,3865,1681],{},[472,3867,708],{"stretchy":702},[418,3869,1686],{"encoding":420},[394,3871,3873],{"className":3872,"ariaHidden":425},[424],[394,3874,3876,3879,3882,3885,3888,3891,3894],{"className":3875},[429],[394,3877],{"className":3878,"style":721},[433],[394,3880,703],{"className":3881},[728],[394,3883,16],{"className":3884},[438,439],[394,3886,1678],{"className":3887},[1705],[394,3889],{"className":3890,"style":1316},[1203],[394,3892,1681],{"className":3893,"style":649},[438,439],[394,3895,708],{"className":3896},[735]," 型张量的分量变换法则为：",[394,3899,3901],{"className":3900,"translate":398},[1100],[394,3902,3904,4147],{"className":3903,"translate":398},[397],[394,3905,3907],{"className":3906},[402],[404,3908,3909],{"xmlns":406,"display":1109},[408,3910,3911,4144],{},[411,3912,3913,3957,3959,3995,3997,4031,4033,4068,4070,4104,4106],{},[3914,3915,3916,3918,3937],"msubsup",{},[414,3917,1730],{},[411,3919,3920,3923,3929,3931],{},[2324,3921,3922],{},"  ",[3257,3924,3925,3927],{},[414,3926,3269],{},[806,3928,970],{},[472,3930,1846],{},[3257,3932,3933,3935],{},[414,3934,3269],{},[414,3936,1681],{},[411,3938,3939,3941],{},[472,3940,3800],{"mathvariant":1424},[411,3942,3943,3949,3951],{},[3257,3944,3945,3947],{},[414,3946,3266],{},[806,3948,970],{},[472,3950,1846],{},[3257,3952,3953,3955],{},[414,3954,3266],{},[414,3956,16],{},[472,3958,1124],{},[1507,3960,3961,3980],{},[411,3962,3963,3966],{},[414,3964,3965],{"mathvariant":1424},"∂",[467,3967,3968,3970],{},[414,3969,3765],{},[411,3971,3972,3974],{},[472,3973,3800],{"mathvariant":1424},[3257,3975,3976,3978],{},[414,3977,3266],{},[806,3979,970],{},[411,3981,3982,3984],{},[414,3983,3965],{"mathvariant":1424},[467,3985,3986,3988],{},[414,3987,3765],{},[3257,3989,3990,3993],{},[414,3991,3992],{},"k",[806,3994,970],{},[472,3996,1846],{},[1507,3998,3999,4017],{},[411,4000,4001,4003],{},[414,4002,3965],{"mathvariant":1424},[467,4004,4005,4007],{},[414,4006,3765],{},[411,4008,4009,4011],{},[472,4010,3800],{"mathvariant":1424},[3257,4012,4013,4015],{},[414,4014,3266],{},[414,4016,16],{},[411,4018,4019,4021],{},[414,4020,3965],{"mathvariant":1424},[467,4022,4023,4025],{},[414,4024,3765],{},[3257,4026,4027,4029],{},[414,4028,3992],{},[414,4030,16],{},[472,4032,2649],{},[1507,4034,4035,4050],{},[411,4036,4037,4039],{},[414,4038,3965],{"mathvariant":1424},[467,4040,4041,4043],{},[414,4042,3765],{},[3257,4044,4045,4048],{},[414,4046,4047],{},"l",[806,4049,970],{},[411,4051,4052,4054],{},[414,4053,3965],{"mathvariant":1424},[467,4055,4056,4058],{},[414,4057,3765],{},[411,4059,4060,4062],{},[472,4061,3800],{"mathvariant":1424},[3257,4063,4064,4066],{},[414,4065,3269],{},[806,4067,970],{},[472,4069,1846],{},[1507,4071,4072,4086],{},[411,4073,4074,4076],{},[414,4075,3965],{"mathvariant":1424},[467,4077,4078,4080],{},[414,4079,3765],{},[3257,4081,4082,4084],{},[414,4083,4047],{},[414,4085,1681],{},[411,4087,4088,4090],{},[414,4089,3965],{"mathvariant":1424},[467,4091,4092,4094],{},[414,4093,3765],{},[411,4095,4096,4098],{},[472,4097,3800],{"mathvariant":1424},[3257,4099,4100,4102],{},[414,4101,3269],{},[414,4103,1681],{},[472,4105,2649],{},[3914,4107,4108,4110,4128],{},[414,4109,1730],{},[411,4111,4112,4114,4120,4122],{},[2324,4113,3922],{},[3257,4115,4116,4118],{},[414,4117,4047],{},[806,4119,970],{},[472,4121,1846],{},[3257,4123,4124,4126],{},[414,4125,4047],{},[414,4127,1681],{},[411,4129,4130,4136,4138],{},[3257,4131,4132,4134],{},[414,4133,3992],{},[806,4135,970],{},[472,4137,1846],{},[3257,4139,4140,4142],{},[414,4141,3992],{},[414,4143,16],{},[418,4145,4146],{"encoding":420},"T'^{i_1 \\cdots i_p}_{\\ \\ j_1 \\cdots j_q} = \\frac{\\partial x'^{i_1}}{\\partial x^{k_1}} \\cdots \\frac{\\partial x'^{i_p}}{\\partial x^{k_p}} \\cdot \\frac{\\partial x^{l_1}}{\\partial x'^{j_1}} \\cdots \\frac{\\partial x^{l_q}}{\\partial x'^{j_q}} \\cdot T^{k_1 \\cdots k_p}_{\\ \\ l_1 \\cdots l_q}",[394,4148,4150,4414,4863,5306],{"className":4149,"ariaHidden":425},[424],[394,4151,4153,4157,4405,4408,4411],{"className":4152},[429],[394,4154],{"className":4155,"style":4156},[433],"height:1.478em;vertical-align:-0.4742em;",[394,4158,4160,4163],{"className":4159},[438],[394,4161,1730],{"className":4162,"style":1745},[438,439],[394,4164,4166],{"className":4165},[497],[394,4167,4169,4396],{"className":4168},[501,860],[394,4170,4172,4393],{"className":4171},[505],[394,4173,4176,4291],{"className":4174,"style":4175},[509],"height:1.0038em;",[394,4177,4179,4182],{"style":4178},"top:-2.4231em;margin-left:-0.1389em;margin-right:0.05em;",[394,4180],{"className":4181,"style":517},[516],[394,4183,4185],{"className":4184},[521,522,523,524],[394,4186,4188,4195,4201,4246,4249],{"className":4187},[438,524],[394,4189,4191],{"className":4190},[1203,524],[394,4192,4194],{"className":4193},[524]," ",[394,4196,4198],{"className":4197},[1203,524],[394,4199,4194],{"className":4200},[524],[394,4202,4204,4207],{"className":4203},[438,524],[394,4205,3269],{"className":4206,"style":3317},[438,439,524],[394,4208,4210],{"className":4209},[497],[394,4211,4213,4237],{"className":4212},[501,860],[394,4214,4216,4234],{"className":4215},[505],[394,4217,4220],{"className":4218,"style":4219},[509],"height:0.3173em;",[394,4221,4223,4227],{"style":4222},"top:-2.357em;margin-left:-0.0572em;margin-right:0.0714em;",[394,4224],{"className":4225,"style":4226},[516],"height:2.5em;",[394,4228,4231],{"className":4229},[521,4230,1002,524],"reset-size3",[394,4232,970],{"className":4233},[438,524],[394,4235,897],{"className":4236},[896],[394,4238,4240],{"className":4239},[505],[394,4241,4244],{"className":4242,"style":4243},[509],"height:0.143em;",[394,4245],{},[394,4247,1846],{"className":4248},[837,524],[394,4250,4252,4255],{"className":4251},[438,524],[394,4253,3269],{"className":4254,"style":3317},[438,439,524],[394,4256,4258],{"className":4257},[497],[394,4259,4261,4282],{"className":4260},[501,860],[394,4262,4264,4279],{"className":4263},[505],[394,4265,4268],{"className":4266,"style":4267},[509],"height:0.1645em;",[394,4269,4270,4273],{"style":4222},[394,4271],{"className":4272,"style":4226},[516],[394,4274,4276],{"className":4275},[521,4230,1002,524],[394,4277,1681],{"className":4278,"style":649},[438,439,524],[394,4280,897],{"className":4281},[896],[394,4283,4285],{"className":4284},[505],[394,4286,4289],{"className":4287,"style":4288},[509],"height:0.2819em;",[394,4290],{},[394,4292,4294,4297],{"style":4293},"top:-3.2421em;margin-right:0.05em;",[394,4295],{"className":4296,"style":517},[516],[394,4298,4300],{"className":4299},[521,522,523,524],[394,4301,4303,4306],{"className":4302},[438,524],[394,4304,3800],{"className":4305},[438,524],[394,4307,4309,4350,4353],{"className":4308},[438,524],[394,4310,4312,4315],{"className":4311},[438,524],[394,4313,3266],{"className":4314},[438,439,524],[394,4316,4318],{"className":4317},[497],[394,4319,4321,4342],{"className":4320},[501,860],[394,4322,4324,4339],{"className":4323},[505],[394,4325,4327],{"className":4326,"style":4219},[509],[394,4328,4330,4333],{"style":4329},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[394,4331],{"className":4332,"style":4226},[516],[394,4334,4336],{"className":4335},[521,4230,1002,524],[394,4337,970],{"className":4338},[438,524],[394,4340,897],{"className":4341},[896],[394,4343,4345],{"className":4344},[505],[394,4346,4348],{"className":4347,"style":4243},[509],[394,4349],{},[394,4351,1846],{"className":4352},[837,524],[394,4354,4356,4359],{"className":4355},[438,524],[394,4357,3266],{"className":4358},[438,439,524],[394,4360,4362],{"className":4361},[497],[394,4363,4365,4385],{"className":4364},[501,860],[394,4366,4368,4382],{"className":4367},[505],[394,4369,4371],{"className":4370,"style":4267},[509],[394,4372,4373,4376],{"style":4329},[394,4374],{"className":4375,"style":4226},[516],[394,4377,4379],{"className":4378},[521,4230,1002,524],[394,4380,16],{"className":4381},[438,439,524],[394,4383,897],{"className":4384},[896],[394,4386,4388],{"className":4387},[505],[394,4389,4391],{"className":4390,"style":4288},[509],[394,4392],{},[394,4394,897],{"className":4395},[896],[394,4397,4399],{"className":4398},[505],[394,4400,4403],{"className":4401,"style":4402},[509],"height:0.4742em;",[394,4404],{},[394,4406],{"className":4407,"style":1204},[1203],[394,4409,1124],{"className":4410},[1208],[394,4412],{"className":4413,"style":1204},[1203],[394,4415,4417,4421,4637,4640,4643,4646,4854,4857,4860],{"className":4416},[429],[394,4418],{"className":4419,"style":4420},[433],"height:2.1877em;vertical-align:-0.686em;",[394,4422,4424,4427,4634],{"className":4423},[438],[394,4425],{"className":4426},[728,1555],[394,4428,4430],{"className":4429},[1507],[394,4431,4433,4625],{"className":4432},[501,860],[394,4434,4436,4622],{"className":4435},[505],[394,4437,4440,4526,4534],{"className":4438,"style":4439},[509],"height:1.5017em;",[394,4441,4443,4446],{"style":4442},"top:-2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x}",[394,5587,5589],{"className":5588,"ariaHidden":425},[424],[394,5590,5592,5596],{"className":5591},[429],[394,5593],{"className":5594,"style":5595},[433],"height:1.3185em;vertical-align:-0.345em;",[394,5597,5599,5602,5699],{"className":5598},[438],[394,5600],{"className":5601},[728,1555],[394,5603,5605],{"className":5604},[1507],[394,5606,5608,5691],{"className":5607},[501,860],[394,5609,5611,5688],{"className":5610},[505],[394,5612,5615,5632,5640],{"className":5613,"style":5614},[509],"height:0.9735em;",[394,5616,5617,5620],{"style":1571},[394,5618],{"className":5619,"style":874},[516],[394,5621,5623],{"className":5622},[521,522,523,524],[394,5624,5626,5629],{"className":5625},[438,524],[394,5627,3965],{"className":5628,"style":4452},[438,524],[394,5630,3765],{"className":5631},[438,439,524],[394,5633,5634,5637],{"style":1586},[394,5635],{"className":5636,"style":874},[516],[394,5638],{"className":5639,"style":1594},[1593],[394,5641,5642,5645],{"style":1597},[394,5643],{"className":5644,"style":874},[516],[394,5646,5648],{"className":5647},[521,522,523,524],[394,5649,5651,5654],{"className":5650},[438,524],[394,5652,3965],{"className":5653,"style":4452},[438,524],[394,5655,5657,5660],{"className":5656},[438,524],[394,5658,3765],{"className":5659},[438,439,524],[394,5661,5663],{"className":5662},[497],[394,5664,5666],{"className":5665},[501],[394,5667,5669],{"className":5668},[505],[394,5670,5673],{"className":5671,"style":5672},[509],"height:0.8278em;",[394,5674,5676,5679],{"style":5675},"top:-2.931em;margin-right:0.0714em;",[394,5677],{"className":5678,"style":4226},[516],[394,5680,5682],{"className":5681},[521,4230,1002,524],[394,5683,5685],{"className":5684},[438,524],[394,5686,3800],{"className":5687},[438,524],[394,5689,897],{"className":5690},[896],[394,5692,5694],{"className":5693},[505],[394,5695,5697],{"className":5696,"style":1619},[509],[394,5698],{},[394,5700],{"className":5701},[735,1555]," 的雅可比因子；",[100,5704,5705,5706,5855],{},"每个下标（对应对偶向量槽位的「协变」分量）乘以一个形如 ",[394,5707,5709,5739],{"className":5708,"translate":398},[397],[394,5710,5712],{"className":5711},[402],[404,5713,5714],{"xmlns":406},[408,5715,5716,5736],{},[411,5717,5718],{},[1507,5719,5720,5726],{},[411,5721,5722,5724],{},[414,5723,3965],{"mathvariant":1424},[414,5725,3765],{},[411,5727,5728,5730],{},[414,5729,3965],{"mathvariant":1424},[467,5731,5732,5734],{},[414,5733,3765],{},[472,5735,3800],{"mathvariant":1424,"lspace":3799,"rspace":3799},[418,5737,5738],{"encoding":420},"\\frac{\\partial x}{\\partial x'}",[394,5740,5742],{"className":5741,"ariaHidden":425},[424],[394,5743,5745,5749],{"className":5744},[429],[394,5746],{"className":5747,"style":5748},[433],"height:1.2251em;vertical-align:-0.345em;",[394,5750,5752,5755,5852],{"className":5751},[438],[394,5753],{"className":5754},[728,1555],[394,5756,5758],{"className":5757},[1507],[394,5759,5761,5844],{"className":5760},[501,860],[394,5762,5764,5841],{"className":5763},[505],[394,5765,5768,5816,5824],{"className":5766,"style":5767},[509],"height:0.8801em;",[394,5769,5770,5773],{"style":1571},[394,5771],{"className":5772,"style":874},[516],[394,5774,5776],{"className":5775},[521,522,523,524],[394,5777,5779,5782],{"className":5778},[438,524],[394,5780,3965],{"className":5781,"style":4452},[438,524],[394,5783,5785,5788],{"className":5784},[438,524],[394,5786,3765],{"className":5787},[438,439,524],[394,5789,5791],{"className":5790},[497],[394,5792,5794],{"className":5793},[501],[394,5795,5797],{"className":5796},[505],[394,5798,5801],{"className":5799,"style":5800},[509],"height:0.6828em;",[394,5802,5804,5807],{"style":5803},"top:-2.786em;margin-right:0.0714em;",[394,5805],{"className":5806,"style":4226},[516],[394,5808,5810],{"className":5809},[521,4230,1002,524],[394,5811,5813],{"className":5812},[438,524],[394,5814,3800],{"className":5815},[438,524],[394,5817,5818,5821],{"style":1586},[394,5819],{"className":5820,"style":874},[516],[394,5822],{"className":5823,"style":1594},[1593],[394,5825,5826,5829],{"style":1597},[394,5827],{"className":5828,"style":874},[516],[394,5830,5832],{"className":5831},[521,522,523,524],[394,5833,5835,5838],{"className":5834},[438,524],[394,5836,3965],{"className":5837,"style":4452},[438,524],[394,5839,3765],{"className":5840},[438,439,524],[394,5842,897],{"className":5843},[896],[394,5845,5847],{"className":5846},[505],[394,5848,5850],{"className":5849,"style":1619},[509],[394,5851],{},[394,5853],{"className":5854},[735,1555]," 的雅可比因子（正是前者的倒数）。",[16,5857,5858],{},"这就是第 2 章那个直观例子「厘米与米」的严格数学版：上下指标的雅可比因子互为倒数，配对缩并时精确抵消，从而保证张量整体（作为几何对象）不变，只有相对于特定坐标系的分量在变。",[11,5860,5861],{"id":5861},"常见张量",[16,5863,5864],{},"下面列举各个领域中遇到的张量示例。",[5866,5867,5868,5946],"table",{},[5869,5870,5871],"thead",{},[5872,5873,5874,5878,5909,5940,5943],"tr",{},[5875,5876,5877],"th",{},"张量类型",[5875,5879,5880,5908],{},[394,5881,5883,5896],{"className":5882,"translate":398},[397],[394,5884,5886],{"className":5885},[402],[404,5887,5888],{"xmlns":406},[408,5889,5890,5894],{},[411,5891,5892],{},[414,5893,16],{},[418,5895,16],{"encoding":420},[394,5897,5899],{"className":5898,"ariaHidden":425},[424],[394,5900,5902,5905],{"className":5901},[429],[394,5903],{"className":5904,"style":1772},[433],[394,5906,16],{"className":5907},[438,439],"（对偶向量槽位）",[5875,5910,5911,5939],{},[394,5912,5914,5927],{"className":5913,"translate":398},[397],[394,5915,5917],{"className":5916},[402],[404,5918,5919],{"xmlns":406},[408,5920,5921,5925],{},[411,5922,5923],{},[414,5924,1681],{},[418,5926,1681],{"encoding":420},[394,5928,5930],{"className":5929,"ariaHidden":425},[424],[394,5931,5933,5936],{"className":5932},[429],[394,5934],{"className":5935,"style":1772},[433],[394,5937,1681],{"className":5938,"style":649},[438,439],"（向量槽位）",[5875,5941,5942],{},"总阶数",[5875,5944,5945],{},"典型例子",[5947,5948,5949,5964,5978,5992,6006,6237],"tbody",{},[5872,5950,5951,5955,5957,5959,5961],{},[5952,5953,5954],"td",{},"标量",[5952,5956,804],{},[5952,5958,804],{},[5952,5960,804],{},[5952,5962,5963],{},"温度、质量",[5872,5965,5966,5969,5971,5973,5975],{},[5952,5967,5968],{},"向量",[5952,5970,804],{},[5952,5972,970],{},[5952,5974,970],{},[5952,5976,5977],{},"速度、力",[5872,5979,5980,5983,5985,5987,5989],{},[5952,5981,5982],{},"对偶向量（1-形式）",[5952,5984,970],{},[5952,5986,804],{},[5952,5988,970],{},[5952,5990,5991],{},"梯度、行向量",[5872,5993,5994,5997,5999,6001,6003],{},[5952,5995,5996],{},"线性变换\u002F惯性张量",[5952,5998,970],{},[5952,6000,970],{},[5952,6002,808],{},[5952,6004,6005],{},"转动惯量、应力张量",[5872,6007,6008,6010,6012,6014,6016],{},[5952,6009,3237],{},[5952,6011,804],{},[5952,6013,808],{},[5952,6015,808],{},[5952,6017,6018,6019],{},"长度定义 ",[394,6020,6022,6072],{"className":6021,"translate":398},[397],[394,6023,6025],{"className":6024},[402],[404,6026,6027],{"xmlns":406},[408,6028,6029,6069],{},[411,6030,6031,6034,6041,6043,6053,6055,6061,6063],{},[414,6032,6033],{},"d",[467,6035,6036,6039],{},[414,6037,6038],{},"s",[806,6040,808],{},[472,6042,1124],{},[3257,6044,6045,6047],{},[414,6046,3261],{},[411,6048,6049,6051],{},[414,6050,3266],{},[414,6052,3269],{},[414,6054,6033],{},[467,6056,6057,6059],{},[414,6058,3765],{},[414,6060,3266],{},[414,6062,6033],{},[467,6064,6065,6067],{},[414,6066,3765],{},[414,6068,3269],{},[418,6070,6071],{"encoding":420},"ds^2 = g_{ij}dx^i dx^j",[394,6073,6075,6123],{"className":6074,"ariaHidden":425},[424],[394,6076,6078,6082,6085,6114,6117,6120],{"className":6077},[429],[394,6079],{"className":6080,"style":6081},[433],"height:0.8141em;",[394,6083,6033],{"className":6084},[438,439],[394,6086,6088,6091],{"className":6087},[438],[394,6089,6038],{"className":6090},[438,439],[394,6092,6094],{"className":6093},[497],[394,6095,6097],{"className":6096},[501],[394,6098,6100],{"className":6099},[505],[394,6101,6103],{"className":6102,"style":6081},[509],[394,6104,6105,6108],{"style":512},[394,6106],{"className":6107,"style":517},[516],[394,6109,6111],{"className":6110},[521,522,523,524],[394,6112,808],{"className":6113},[438,524],[394,6115],{"className":6116,"style":1204},[1203],[394,6118,1124],{"className":6119},[1208],[394,6121],{"className":6122,"style":1204},[1203],[394,6124,6126,6130,6173,6176,6205,6208],{"className":6125},[429],[394,6127],{"className":6128,"style":6129},[433],"height:1.1108em;vertical-align:-0.2861em;",[394,6131,6133,6136],{"className":6132},[438],[394,6134,3261],{"className":6135,"style":649},[438,439],[394,6137,6139],{"className":6138},[497],[394,6140,6142,6165],{"className":6141},[501,860],[394,6143,6145,6162],{"className":6144},[505],[394,6146,6148],{"className":6147,"style":3301},[509],[394,6149,6150,6153],{"style":3304},[394,6151],{"className":6152,"style":517},[516],[394,6154,6156],{"className":6155},[521,522,523,524],[394,6157,6159],{"className":6158},[438,524],[394,6160,3318],{"className":6161,"style":3317},[438,439,524],[394,6163,897],{"className":6164},[896],[394,6166,6168],{"className":6167},[505],[394,6169,6171],{"className":6170,"style":3328},[509],[394,6172],{},[394,6174,6033],{"className":6175},[438,439],[394,6177,6179,6182],{"className":6178},[438],[394,6180,3765],{"className":6181},[438,439],[394,6183,6185],{"className":6184},[497],[394,6186,6188],{"className":6187},[501],[394,6189,6191],{"className":6190},[505],[394,6192,6194],{"className":6193,"style":4564},[509],[394,6195,6196,6199],{"style":512},[394,6197],{"className":6198,"style":517},[516],[394,6200,6202],{"className":6201},[521,522,523,524],[394,6203,3266],{"className":6204},[438,439,524],[394,6206,6033],{"className":6207},[438,439],[394,6209,6211,6214],{"className":6210},[438],[394,6212,3765],{"className":6213},[438,439],[394,6215,6217],{"className":6216},[497],[394,6218,6220],{"className":6219},[501],[394,6221,6223],{"className":6222},[505],[394,6224,6226],{"className":6225,"style":4564},[509],[394,6227,6228,6231],{"style":512},[394,6229],{"className":6230,"style":517},[516],[394,6232,6234],{"className":6233},[521,522,523,524],[394,6235,3269],{"className":6236,"style":3317},[438,439,524],[5872,6238,6239,6242,6244,6246,6248],{},[5952,6240,6241],{},"黎曼曲率张量",[5952,6243,970],{},[5952,6245,817],{},[5952,6247,977],{},[5952,6249,6250],{},"广义相对论中的时空弯曲",[11,6252,6253],{"id":6253},"两种语境",[16,6255,6256],{},"最后需要澄清一点：「张量」一词在深度学习和在物理\u002F微分几何中，指的是两个不同抽象层级上的概念，这是困惑的重要来源。",[97,6258,6259,6266],{},[100,6260,6261,6262,6265],{},"深度学习：",[375,6263,6264],{},"tensor"," 只是一个多维数组；唯一关心的是并行计算的效率。这里没有「坐标变换不变性」，也没有协变与逆变之分。把张量理解成「广义数组」就完全够用，不必给这个词背上哲学重负。",[100,6267,6268],{},"物理与微分几何：张量首先是一个不依赖任何坐标选择的几何或物理对象；数组只是它相对于某个坐标基的「临时快照」。协变、逆变、度量、缩并——这些概念存在的目的只有一个：保证从这些临时快照算出的物理量（长度、能量、曲率）是客观的、绝对的，不随观察者改变。",[16,6270,6271],{},"两种语境没有对错之分；它们服务的目的大相径庭。真正的困惑往往来自用深度学习的直觉去理解广义相对论的张量，或者反过来。一旦意识到这是两回事，很多「教科书讲不清楚」式的挫败感就会自行消解。",[11,6273,6274],{"id":6274},"结语",[16,6276,6277],{},"让我们回到最初的问题：张量到底是什么？",[16,6279,6280],{},"它是一台带有若干「槽位」的机器：有的槽位等待「箭头」（向量），有的槽位等待「标尺」（对偶向量）。当所有槽位都被填满时，它输出的数字永远相同——无论你从哪个坐标系观察，无论你用什么单位度量。而度量张量，则是连接向量世界与对偶向量世界的桥梁，使「长度」这样一个看似无辜的概念得以被严格定义。",[16,6282,6283],{},"下次你写下下标或上标时，不妨停下来想一想：这个位置是在等一支箭头，还是在等一把标尺？",{"title":322,"searchDepth":323,"depth":323,"links":6285},[6286,6287,6288,6289,6290,6291,6292,6293,6294],{"id":367,"depth":323,"text":367},{"id":444,"depth":323,"text":444},{"id":1652,"depth":323,"text":1652},{"id":2398,"depth":323,"text":2398},{"id":3237,"depth":323,"text":3237},{"id":3747,"depth":323,"text":3747},{"id":5861,"depth":323,"text":5861},{"id":6253,"depth":323,"text":6253},{"id":6274,"depth":323,"text":6274},{},"2026-05-30","\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor",{"title":345,"description":322},"blog\u002F2026\u002F2026-05-30-about-the-tensor","向量是箭头，对偶向量是标尺——它们的配对产生不变的标量。张量是带多个可填充槽位的线性机器，化解了「向量进向量出」与「输出标量」之间的悖论。度量张量连接二者，使长度与能量保持坐标不变。",[6302,6303],"mathematics","machine-learning","XKmR1vLRKY6-cwYBkWzJW8HOZ_L_jhd1AjH3wujbMCU",{"id":6306,"title":6307,"body":6308,"description":322,"draft":330,"enableComment":334,"extension":331,"image":322,"important":330,"location":332,"meta":6346,"navigation":334,"ogImage":332,"onday":6347,"path":6348,"seo":6349,"stem":6350,"summary":6351,"tags":6352,"__hash__":6353},"blog\u002Fblog\u002F2026\u002F2026-05-13-the-pale-blue-dot.md","暗淡蓝点",{"type":8,"value":6309,"toc":6344},[6310,6315,6318,6321,6324,6327,6330,6333,6336,6339],[16,6311,6312],{},[48,6313],{"alt":322,"src":6314},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202605132235273.png",[16,6316,6317],{},"再看看那个小点。就在这里。就是家。就是我们。",[16,6319,6320],{},"所有你爱的人，所有你认识的人，所有你听说过的人，所有曾经存在过的人类，都在上面度过了他们的一生。",[16,6322,6323],{},"我们欢乐和痛苦的总和，成千上万自信的宗教、意识形态和经济学说，每一个猎人和觅食者，每一个英雄和懦夫，每一个文明的创造者和毁灭者，每一个国王和农民，每一对相爱的情侣，每一位母亲和父亲，充满希望的孩子，发明家和探险家，每一个道德教师，每一个腐败的政客，每一个“超级巨星”，每一个“最高领导人”，每一个圣人和罪人，都在那里生活过——在一个悬浮在阳光中的尘埃微粒上。",[16,6325,6326],{},"地球在浩瀚的宇宙舞台上只是一个小小的舞台。想想所有那些将军和皇帝流淌的血河，为了荣耀和胜利，他们可以成为一个小点的一小部分的瞬间主人。",[16,6328,6329],{},"想想这个像素的一个角落的居民对另一个几乎无法区分的角落的居民所施加的无休止的残忍，他们之间的误解是多么频繁，他们是多么渴望互相残杀，他们的仇恨是多么强烈。",[16,6331,6332],{},"我们的姿态，我们想象的自命不凡，我们认为我们在宇宙中拥有某种特权地位的错觉，都受到了这个苍白的光点的挑战。我们的星球是浩瀚的宇宙黑暗中孤独的一点。",[16,6334,6335],{},"在我们的默默无闻中，在所有这些广阔的空间中，没有任何迹象表明会有来自其他地方的帮助来拯救我们自己。到目前为止，地球是唯一已知孕育生命的星球。至少在不久的将来，我们的物种无处可去。参观，可以。定居，还不行。不管你喜欢与否，目前地球就是我们站立的地方。",[16,6337,6338],{},"有人说，天文学是一种令人谦卑和塑造性格的体验。对于人类自负的愚蠢，也许没有比我们这个小世界的遥远图像更好的证明了。对我来说，它强调了我们彼此更友善地相处，并保护和珍惜这个苍白蓝点的责任，这是我们唯一知道的家园。",[32,6340,6341],{},[16,6342,6343],{},"所有的战争与和平\n爱恨与悲欢\n都在这颗名为「地球」的微尘上\n日复一日地上演。",{"title":322,"searchDepth":323,"depth":323,"links":6345},[],{},"2026-05-13","\u002Fblog\u002F2026\u002F2026-05-13-the-pale-blue-dot",{"title":6307,"description":322},"blog\u002F2026\u002F2026-05-13-the-pale-blue-dot","所有的战争与和平，爱恨与悲欢，都在这颗名为「地球」的微尘上，日复一日地上演。",[341],"fVZwpQuTB11Vpr_Nh2S5mrdM3Z98pnu5QfBsLvO025o",1787145928663]