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href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%BC%B5%E9%87%8F&returntoquery=variant%3Dzh-hans" title="我们推荐您登录，但这不是强制性的[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" lang="zh-Hans" dir="ltr"> <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="有关本IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-歷史背景" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#歷史背景"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>历史背景</span> </div> </a> <ul id="toc-歷史背景-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>定义</span> </div> </a> <ul id="toc-定义-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-方法的选择" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#方法的选择"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>方法的选择</span> </div> </a> <ul id="toc-方法的选择-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-方法细节" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#方法细节"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>方法细节</span> </div> </a> <ul id="toc-方法细节-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-张量密度" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#张量密度"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>张量密度</span> </div> </a> <ul id="toc-张量密度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-張量阶" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#張量阶"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>张量阶</span> </div> </a> <ul id="toc-張量阶-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参閱" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参閱"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>参阅</span> </div> </a> <button aria-controls="toc-参閱-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关参阅子章节</span> </button> <ul id="toc-参閱-sublist" class="vector-toc-list"> <li id="toc-记法常规" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#记法常规"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>记法常规</span> </div> </a> <ul id="toc-记法常规-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-基础" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#基础"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>基础</span> </div> </a> <ul id="toc-基础-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-应用" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#应用"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>应用</span> </div> </a> <ul id="toc-应用-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-参考资料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>参考资料</span> </div> </a> <ul id="toc-参考资料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考書籍" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考書籍"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>参考书籍</span> </div> </a> <ul id="toc-參考書籍-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>外部链接</span> </div> </a> <button aria-controls="toc-外部链接-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关外部链接子章节</span> </button> <ul id="toc-外部链接-sublist" class="vector-toc-list"> <li id="toc-张量软件" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#张量软件"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>张量软件</span> </div> </a> <ul id="toc-张量软件-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" lang="zh-Hans" dir="ltr"> <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading" lang="zh-Hans" dir="ltr"><span class="mw-page-title-main">张量</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" lang="zh-Hans" dir="ltr"> <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。60种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-60" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">60种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%88%D8%AA%D8%B1" title="موتر – 阿拉伯语" lang="ar" hreflang="ar" data-title="موتر" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Tenzor" title="Tenzor – 阿塞拜疆语" lang="az" hreflang="az" data-title="Tenzor" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 保加利亚语" lang="bg" hreflang="bg" data-title="Тензор" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9F%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%B8%E0%A6%B0" title="টেন্সর – 孟加拉语" lang="bn" hreflang="bn" data-title="টেন্সর" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tensor" title="Tensor – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Tensor" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Tenzor" title="Tenzor – 捷克语" lang="cs" hreflang="cs" data-title="Tenzor" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 楚瓦什语" lang="cv" hreflang="cv" data-title="Тензор" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Tensor" title="Tensor – 德语" lang="de" hreflang="de" data-title="Tensor" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B1%CE%BD%CF%85%CF%83%CF%84%CE%AE%CF%82" title="Τανυστής – 希腊语" lang="el" hreflang="el" data-title="Τανυστής" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Tensor" title="Tensor – 英语" lang="en" hreflang="en" data-title="Tensor" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tensoro" title="Tensoro – 世界语" lang="eo" hreflang="eo" data-title="Tensoro" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tensor" title="Tensor – 西班牙语" lang="es" hreflang="es" data-title="Tensor" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tensor" title="Tensor – 爱沙尼亚语" lang="et" hreflang="et" data-title="Tensor" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tentsore" title="Tentsore – 巴斯克语" lang="eu" hreflang="eu" data-title="Tentsore" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D9%86%D8%B3%D9%88%D8%B1" title="تانسور – 波斯语" lang="fa" hreflang="fa" data-title="تانسور" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tensori" title="Tensori – 芬兰语" lang="fi" hreflang="fi" data-title="Tensori" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tenseur" title="Tenseur – 法语" lang="fr" hreflang="fr" data-title="Tenseur" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tensor" title="Tensor – 加利西亚语" lang="gl" hreflang="gl" data-title="Tensor" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%A0%D7%96%D7%95%D7%A8" title="טנזור – 希伯来语" lang="he" hreflang="he" data-title="טנזור" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A6%E0%A4%BF%E0%A4%B6" title="प्रदिश – 印地语" lang="hi" hreflang="hi" data-title="प्रदिश" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tenzor" title="Tenzor – 克罗地亚语" lang="hr" hreflang="hr" data-title="Tenzor" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Tenzor" title="Tenzor – 匈牙利语" lang="hu" hreflang="hu" data-title="Tenzor" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A5%D5%B6%D5%A6%D5%B8%D6%80" title="Տենզոր – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Տենզոր" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tensor" title="Tensor – 印度尼西亚语" lang="id" hreflang="id" data-title="Tensor" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tensore" title="Tensore – 意大利语" lang="it" hreflang="it" data-title="Tensore" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB" title="テンソル – 日语" lang="ja" hreflang="ja" data-title="テンソル" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 哈萨克语" lang="kk" hreflang="kk" data-title="Тензор" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%85%90%EC%84%9C" title="텐서 – 韩语" lang="ko" hreflang="ko" data-title="텐서" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Тензор" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Tensor_(mathematica)" title="Tensor (mathematica) – 拉丁语" lang="la" hreflang="la" data-title="Tensor (mathematica)" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tenzorius" title="Tenzorius – 立陶宛语" lang="lt" hreflang="lt" data-title="Tenzorius" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tenzors" title="Tenzors – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Tenzors" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 马其顿语" lang="mk" hreflang="mk" data-title="Тензор" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A6%E0%B4%BF%E0%B4%B6%E0%B4%82" title="പ്രദിശം – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="പ്രദിശം" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tensor" title="Tensor – 马来语" lang="ms" hreflang="ms" data-title="Tensor" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%90%E1%80%AC%E1%80%A1%E1%80%AF%E1%80%B6" title="တာအုံ – 缅甸语" lang="my" hreflang="my" data-title="တာအုံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tensor" title="Tensor – 荷兰语" lang="nl" hreflang="nl" data-title="Tensor" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tensor" title="Tensor – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Tensor" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tensor" title="Tensor – 书面挪威语" lang="nb" hreflang="nb" data-title="Tensor" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9F%E0%A9%88%E0%A8%82%E0%A8%B8%E0%A8%B0" title="ਟੈਂਸਰ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਟੈਂਸਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tensor" title="Tensor – 波兰语" lang="pl" hreflang="pl" data-title="Tensor" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tensor" title="Tensor – 葡萄牙语" lang="pt" hreflang="pt" data-title="Tensor" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tensor" title="Tensor – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Tensor" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 俄语" lang="ru" hreflang="ru" data-title="Тензор" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tenzor" title="Tenzor – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Tenzor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tensor" title="Tensor – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tensor" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Tenzor" title="Tenzor – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Tenzor" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Tenzor" title="Tenzor – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Tenzor" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Trajtimi_klasik_i_tensor%C3%ABve" title="Trajtimi klasik i tensorëve – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Trajtimi klasik i tensorëve" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Тензор" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Tensor" title="Tensor – 瑞典语" lang="sv" hreflang="sv" data-title="Tensor" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Tensor" title="Tensor – 西里西亚语" lang="szl" hreflang="szl" data-title="Tensor" data-language-autonym="Ślůnski" data-language-local-name="西里西亚语" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tensor" title="Tensor – 他加禄语" lang="tl" hreflang="tl" data-title="Tensor" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tens%C3%B6r" title="Tensör – 土耳其语" lang="tr" hreflang="tr" data-title="Tensör" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 鞑靼语" lang="tt" hreflang="tt" data-title="Тензор" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80" title="Тензор – 乌克兰语" lang="uk" hreflang="uk" data-title="Тензор" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tenzor" title="Tenzor – 乌兹别克语" lang="uz" hreflang="uz" data-title="Tenzor" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Tenx%C6%A1" title="Tenxơ – 越南语" lang="vi" hreflang="vi" data-title="Tenxơ" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BC%A0%E9%87%8F" title="张量 – 吴语" lang="wuu" hreflang="wuu" data-title="张量" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BC%B5%E9%87%8F" title="張量 – 粤语" lang="yue" hreflang="yue" data-title="張量" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q188524#sitelinks-wikipedia" title="编辑跨语言链接" 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lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" lang="zh-Hans" dir="ltr"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%BC%B5%E9%87%8F"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=history" title="该页面过去的修订[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown 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accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" lang="zh-Hans" dir="ltr"> <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E5%BC%B5%E9%87%8F" title="所有链接至本页面的wiki页面列表[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E5%BC%B5%E9%87%8F" rel="nofollow" title="链自本页的页面的最近更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" 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href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fzh-hans%2F%25E5%25BC%25B5%25E9%2587%258F"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" lang="zh-Hans" dir="ltr"> <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E5%BC%B5%E9%87%8F&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印版本</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" lang="zh-Hans" dir="ltr"> <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Tensors" hreflang="en"><span>维基共享资源</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q188524" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-5f97c608" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub" lang="zh-Hans" dir="ltr"><div id="mw-content-subtitle" lang="zh-Hans" dir="ltr"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh-Hans" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/25px-Disambig_gray.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/38px-Disambig_gray.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/50px-Disambig_gray.svg.png 2x" data-file-width="220" data-file-height="168" /></a></span><style data-mw-deduplicate="TemplateStyles:r74069148">body:not(.skin-minerva) .mw-parser-output .ifmobile>.mobile{display:none}body.skin-minerva .mw-parser-output .ifmobile>.nomobile{display:inherit;display:initial}</style><span class="ifmobile"><span class="nomobile">  </span><span class="mobile"></span></span>关于与“<b>张量</b>”标题相近或相同的条目页，请见“<b><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F_(%E6%B6%88%E6%AD%A7%E7%BE%A9)&action=edit&redlink=1" class="new" title="张量 (消歧义)（页面不存在）">张量 (消歧义)</a></b>”。</div> <div id="noteTA-5f97c608" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Physics"></div></div></div> <style data-mw-deduplicate="TemplateStyles:r84596247">.mw-parser-output .multiple-issues-text{width:95%;margin:0.2em 0}.mw-parser-output .multiple-issues-text>.mw-collapsible-content{margin-top:0.3em}.mw-parser-output .compact-ambox .ambox{border:none;border-collapse:collapse;background-color:transparent;margin:0 0 0 1.6em!important;padding:0!important;width:auto;display:block}body.mediawiki .mw-parser-output .compact-ambox .ambox.mbox-small-left{font-size:100%;width:auto;margin:0}.mw-parser-output .compact-ambox .ambox .mbox-text{padding:0!important;margin:0!important}.mw-parser-output .compact-ambox .ambox .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:disc}body.skin-minerva .mw-parser-output .multiple-issues-text>.mw-collapsible-toggle,.mw-parser-output .compact-ambox .ambox .mbox-image,.mw-parser-output .compact-ambox .ambox .mbox-imageright,.mw-parser-output .compact-ambox .ambox .mbox-empty-cell{display:none}</style><style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-问题条目 plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>本条目存在以下问题</b>，请协助<b><a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit">改善本条目</a></b>或在<b><a href="/wiki/Talk:%E5%BC%B5%E9%87%8F" title="Talk:张量">讨论页</a></b>针对议题发表看法。 <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Refimprove plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此条目<b>需要补充更多<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2018年4月28日</span>)</i></small><span class="hide-when-compact"><br /><small>请协助补充多方面<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit">改善这篇条目</a>，<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AF%81" class="mw-redirect" title="Wikipedia:可供查证">无法查证</a>的内容可能会因为<a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">异议提出</a>而被移除。<br />致使用者：请搜索一下条目的标题（来源搜索：<span class="plainlinks"><a rel="nofollow" class="external text" href="//www.google.com/search?&as_eq=wikipedia&q=%22%E5%BC%B5%E9%87%8F%22">"张量"</a> — <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22%E5%BC%B5%E9%87%8F%22">网页</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22%E5%BC%B5%E9%87%8F%22">新闻</a>、<a rel="nofollow" class="external text" href="//books.google.com/books?&as_brr=0&as_pub=-icon&q=%22%E5%BC%B5%E9%87%8F%22">书籍</a>、<a rel="nofollow" class="external text" href="//scholar.google.com/scholar?&q=%22%E5%BC%B5%E9%87%8F%22">学术</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=isch&safe=off&q=%22%E5%BC%B5%E9%87%8F%22">图像</a></span>），以检查网络上是否存在该主题的更多可靠来源（<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">判定指引</a>）。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Copy_edit plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Ambox_spelling.svg/48px-Ambox_spelling.svg.png" decoding="async" width="48" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Ambox_spelling.svg/72px-Ambox_spelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Ambox_spelling.svg/96px-Ambox_spelling.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">此条目<b>需要<a href="/wiki/Wikipedia:%E5%A6%82%E4%BD%95%E5%AF%A9%E6%A0%B8%E6%A0%A1%E5%B0%8D" class="mw-redirect" title="Wikipedia:如何审核校对">编修</a>，以确保文法、<a href="/wiki/Wikipedia:%E9%81%BF%E5%85%8D%E4%BD%BF%E7%94%A8%E7%9A%84%E5%AD%97%E8%A9%9E" class="mw-redirect" title="Wikipedia:避免使用的字词">用词、语气</a>、<a href="/wiki/Wikipedia:%E6%A0%BC%E5%BC%8F%E6%89%8B%E5%86%8C" title="Wikipedia:格式手册">格式</a>、<a href="/wiki/Wikipedia:%E6%A0%BC%E5%BC%8F%E6%89%8B%E5%86%8C/%E6%A0%87%E7%82%B9%E7%AC%A6%E5%8F%B7" title="Wikipedia:格式手册/标点符号">标点</a>等使用恰当。</b><span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2014年9月3日</span>)</i></small><span class="hide-when-compact"><br /><small>请按照<a href="/wiki/Wikipedia:%E5%A6%82%E4%BD%95%E5%AF%A9%E6%A0%B8%E6%A0%A1%E5%B0%8D" class="mw-redirect" title="Wikipedia:如何审核校对">校对指引</a>，帮助<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit">编辑</a>这个条目。（<a href="/wiki/Help:%E7%BC%96%E8%BE%91" class="mw-redirect" title="Help:编辑">帮助</a>、<a href="/wiki/Talk:%E5%BC%B5%E9%87%8F" title="Talk:张量">讨论</a>）</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Expert_needed plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">此条目需要<b>精通或熟悉相关主题的编者</b>参与及协助编辑。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2011年3月12日</span>)</i></small><span class="hide-when-compact"><br /><small>请<a href="/wiki/Wikipedia:%E5%B0%88%E5%AE%B6%E9%82%80%E8%AB%8B%E5%B7%A5%E4%BD%9C%E5%B0%8F%E7%B5%84" title="Wikipedia:专家邀请工作小组">邀请</a>适合的人士<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit">改善本条目</a>。更多的细节与详情请参见<a href="/wiki/Talk:%E5%BC%B5%E9%87%8F" title="Talk:张量">讨论页</a>。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> </div></div><span class="hide-when-compact"></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <p><b>张量</b>（英语：<span lang="en">Tensor</span>）在数学中是一个<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AF%B9%E8%B1%A1" title="数学对象">代数对象</a>，描述了与向量空间相关的代数对象集之间的<a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="多重线性映射">多重线性映射</a>。张量可以作为不同的对象之间的<a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a>，例如<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a>、<a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a>，甚至其他张量。张量有很多种类型，包括标量和向量、<a href="/wiki/%E7%B7%9A%E6%80%A7%E6%B3%9B%E5%87%BD" title="线性泛函">对偶向量</a>、<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a>之间的多重线性映射，甚至还有一些运算，例如<a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">点积</a>。张量的定义独立于任何<a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (线性代数)">基</a>，尽管它们通常由与特定坐标系相关的基中的<a href="/wiki/%E5%88%86%E9%87%8F" class="mw-redirect" title="分量">分量</a>来表示；这些分量形成一个<a href="/wiki/%E6%95%B0%E7%BB%84" title="数组">数组</a>，可以将其视为高维<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>维空间上的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>阶张量有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b016a3a6c7f94bf3b634260c24bf8cd81bbf51d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.368ex; height:2.343ex;" alt="{\displaystyle n^{r}}"></span>个分量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>也称为该张量的<a href="/w/index.php?title=%E7%A7%A9_(%E5%BC%A0%E9%87%8F)&action=edit&redlink=1" class="new" title="秩 (张量)（页面不存在）">秩</a>（与矩阵的秩和阶均无关系）。 </p><p>在<a href="/wiki/%E5%90%8C%E6%9E%84" title="同构">同构</a>的意义下，第零阶张量（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span>）为<a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">纯量</a>，第一阶张量（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span>）为<a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a>，第二阶张量（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19b110d7eb52a69381b88554e63c8a2aef376c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=2}"></span>）则成为<a href="/wiki/%E7%9F%A9%E9%99%A3" class="mw-redirect" title="矩阵">矩阵</a>。例如，对于3维空间，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"></span>时的张量为此向量：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x,y,z\right)^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x,y,z\right)^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b069fe9ae3e821cabbaa61124676598dda2f192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.869ex; height:3.343ex;" alt="{\displaystyle \left(x,y,z\right)^{\mathrm {T} }}"></span>。张量不仅仅是由一定数量的分量组成的数组，在<a href="/wiki/%E5%9D%90%E6%A0%87%E5%8F%98%E6%8D%A2" class="mw-redirect" title="坐标变换">坐标变换</a>时，张量的分量也依照某些规则作<a href="/wiki/%E7%BA%BF%E6%80%A7%E5%8F%98%E6%8D%A2" class="mw-redirect" title="线性变换">线性变换</a>。由于变换方式的不同，张量分成“<a href="/wiki/%E5%85%B1%E8%AE%8A%E5%92%8C%E5%8F%8D%E8%AE%8A" title="共变和反变">协变张量</a>”（指标在下者）、“<a href="/wiki/%E5%85%B1%E8%AE%8A%E5%92%8C%E5%8F%8D%E8%AE%8A" title="共变和反变">逆变张量</a>”（指标在上者）、“混合张量”（指标在上和指标在下两者都有）三类。张量的抽象理论是<a href="/wiki/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" class="mw-redirect" title="线性代数">线性代数</a>分支，现在叫做<a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="多重线性代数">多重线性代数</a>。 </p><p>张量在<a href="/wiki/%E7%89%A9%E7%90%86" class="mw-redirect" title="物理">物理</a>和<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%B8" class="mw-redirect" title="工程学">工程学</a>中很重要。例如在<a href="/w/index.php?title=%E6%89%A9%E6%95%A3%E5%BC%A0%E9%87%8F%E6%88%90%E5%83%8F&action=edit&redlink=1" class="new" title="扩散张量成像（页面不存在）">扩散张量成像</a>中，表达器官对于水的在各个方向的微分<a href="/w/index.php?title=%E9%80%8F%E6%80%A7&action=edit&redlink=1" class="new" title="透性（页面不存在）">透性</a>的张量可以用来产生<a href="/wiki/%E5%A4%A7%E8%84%91" title="大脑">大脑</a>的扫描图。工程上的例子有<a href="/wiki/%E5%BA%94%E5%8A%9B%E5%BC%A0%E9%87%8F" class="mw-redirect" title="应力张量">应力张量</a>和<a href="/w/index.php?title=%E5%BA%94%E5%8F%98%E5%BC%A0%E9%87%8F&action=edit&redlink=1" class="new" title="应变张量（页面不存在）">应变张量</a>，它们都是<a href="#张量阶">二阶张量</a>，对于一般线性材料他们之间的关系由一个四阶<a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E5%BC%B9%E6%80%A7&action=edit&redlink=1" class="new" title="线性弹性（页面不存在）">弹性张量</a>来决定。 </p><p>张量在物理学中提供了一个简明的数学框架用来描述和解决力学（<a href="/wiki/%E6%87%89%E5%8A%9B" title="应力">应力</a>、弹性、<a href="/wiki/%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6" title="流体力学">流体力学</a>、<a href="/wiki/%E6%83%AF%E6%80%A7%E7%9F%A9" class="mw-disambig" title="惯性矩">惯性矩</a>等）、<a href="/wiki/%E7%94%B5%E5%8A%A8%E5%8A%9B%E5%AD%A6" class="mw-redirect" title="电动力学">电动力学</a>（电磁张量、麦克斯韦张量、<a href="/wiki/%E4%BB%8B%E7%94%B5%E5%B8%B8%E6%95%B0" class="mw-redirect" title="介电常数">介电常数</a>、<a href="/wiki/%E7%A3%81%E5%8C%96%E7%8E%87" title="磁化率">磁化率</a>等）、<a href="/wiki/%E5%B9%BF%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" class="mw-redirect" title="广义相对论">广义相对论</a>（应力-能量张量、<a href="/wiki/%E6%9B%B2%E7%8E%87%E5%BC%A0%E9%87%8F" class="mw-disambig" title="曲率张量">曲率张量</a>等）物理问题。在应用中，数学家通常会研究在物体的不同点之间的张量变化。例如，一个物体内的应力可能因位置不同而改变。这就引出了<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a>的概念。在某些领域，张量场十分普遍以至于它们通常被简称为“张量”。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="歷史背景"><span id=".E6.AD.B7.E5.8F.B2.E8.83.8C.E6.99.AF"></span>历史背景</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=1" title="编辑章节：历史背景"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>“张量”一词最初由<a href="/wiki/%E5%A8%81%E5%BB%89%C2%B7%E7%BD%97%E6%81%A9%C2%B7%E5%93%88%E5%AF%86%E9%A1%BF" class="mw-redirect" title="威廉·罗恩·哈密顿">威廉·罗恩·哈密顿</a>在1846年引入，但他把这个词用于指代现在称为<a href="/wiki/%E6%A8%A1" title="模">模</a>的对象。该词的现代意义是<a href="/w/index.php?title=%E6%B2%83%E5%B0%94%E5%BE%B7%E9%A9%AC%E5%B0%94%C2%B7%E7%A6%8F%E6%A0%BC%E7%89%B9&action=edit&redlink=1" class="new" title="沃尔德马尔·福格特（页面不存在）">沃尔德马尔·福格特</a>在1899年开始使用的。 </p><p>这个概念由<a href="/wiki/%E6%A0%BC%E9%9B%B7%E6%88%88%E9%87%8C%E5%A5%A5%C2%B7%E9%87%8C%E5%A5%87-%E5%BA%93%E5%B0%94%E5%B7%B4%E6%96%AF%E6%89%98%E7%BD%97" title="格雷戈里奥·里奇-库尔巴斯托罗">格雷戈里奥·里奇-库尔巴斯托罗</a>在1890年在《<a href="/w/index.php?title=%E7%BB%9D%E5%AF%B9%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95&action=edit&redlink=1" class="new" title="绝对微分几何（页面不存在）">绝对微分几何</a>》的标题下发展出来，随着1900年<a href="/wiki/%E5%88%97%E7%B6%AD-%E5%A5%87%E7%B6%AD%E5%A1%94" class="mw-redirect" title="列维-奇维塔">列维-奇维塔</a>的经典文章《<a href="/wiki/%E7%BB%9D%E5%AF%B9%E5%BE%AE%E5%88%86" class="mw-redirect" title="绝对微分">绝对微分</a>》（意大利文，随后出版了其他译本）的出版而为许多数学家所知。随着1915年左右<a href="/wiki/%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6" class="mw-redirect" title="爱因斯坦">爱因斯坦</a>的<a href="/wiki/%E5%B9%BF%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" class="mw-redirect" title="广义相对论">广义相对论</a>的引入，张量微积分获得了更广泛的承认。广义相对论完全由张量语言表述，爱因斯坦从列维-奇维塔本人那里学了很多张量语言（其实是Marcel Grossman，他是爱因斯坦在<a href="/wiki/%E8%8B%8F%E9%BB%8E%E4%B8%96%E8%81%94%E9%82%A6%E7%90%86%E5%B7%A5%E5%AD%A6%E9%99%A2" title="苏黎世联邦理工学院">苏黎世联邦理工学院</a>的同学，一个几何学家，也是爱因斯坦在张量语言方面的良师益友 - 参看Abraham Pais所著《上帝是微妙的（Subtle is the Lord）》），并学得很艰苦。但张量也用于其它领域，例如<a href="/w/index.php?title=%E8%BF%9E%E7%BB%AD%E5%8A%9B%E5%AD%A6&action=edit&redlink=1" class="new" title="连续力学（页面不存在）">连续力学</a>，譬如<a href="/w/index.php?title=%E5%BA%94%E5%8F%98%E5%BC%A0%E9%87%8F&action=edit&redlink=1" class="new" title="应变张量（页面不存在）">应变张量</a>（参看<a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E5%BC%B9%E6%80%A7&action=edit&redlink=1" class="new" title="线性弹性（页面不存在）">线性弹性</a>）。 </p><p>注意“张量”一词经常用作<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a>的简写，而张量场是对<a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a>的每一点给定一个张量值。要更好的理解张量场，必须首先理解张量的基本思想。 </p> <div class="mw-heading mw-heading2"><h2 id="定义"><span id=".E5.AE.9A.E4.B9.89"></span>定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=2" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个 <style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">(p,q)</span></span>型的张量 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml mvar" style="font-style:italic;margin-left:2px;margin-right:2px;">T</span></span> 被定义为一个多重线性映射（英语：<span lang="en">multilinear map</span>）<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ 個}}}\times \underbrace {V\times \dots \times V} _{q{\text{ 個}}}\mapsto \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> 個</mtext> </mrow> </mrow> </munder> <mo>×</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>V</mi> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>V</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> 個</mtext> </mrow> </mrow> </munder> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ 個}}}\times \underbrace {V\times \dots \times V} _{q{\text{ 個}}}\mapsto \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb47db3c4fcdedf1f075feb983d98fdad0f9d1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:38.678ex; height:6.676ex;" alt="{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ 個}}}\times \underbrace {V\times \dots \times V} _{q{\text{ 個}}}\mapsto \mathbb {R} ,}"></span></dd></dl> <p>其中 <i>V</i> 是<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a>，<i>V</i> <sup>∗</sup> 是其<a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="方法的选择"><span id=".E6.96.B9.E6.B3.95.E7.9A.84.E9.80.89.E6.8B.A9"></span>方法的选择</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=3" title="编辑章节：方法的选择"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>有两种定义张量的方法： </p> <ul><li>通常定义张量的物理学或传统数学方法，是把张量看成一个多维数组，当变换座标或变换基底时，其分量会按照一定变换的规则，这些规则有两种：即<a href="/wiki/%E5%8D%8F%E5%8F%98" class="mw-redirect mw-disambig" title="协变">协变</a>或<a href="/wiki/%E9%80%86%E5%8F%98" class="mw-redirect mw-disambig" title="逆变">逆变</a>转换。</li> <li>通常现代数学中的方法，是把张量定义成某个向量空间或其对偶空间上的多重线性映射，这<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a>在需要引入基底之前不固定任何坐标系统。例如协变向量，可以描述为<a href="/wiki/1-%E5%BD%A2%E5%BC%8F" title="1-形式">1-形式</a>，或者作为逆变向量的<a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a>的元素。</li></ul> <p>但物理学家和工程师是首先识别出向量和张量作为实体具有物理上的意义的，它超越了它们的分量所被表述的（经常是任意的）坐标系。同样，数学家发现有一些张量关系在坐标表示中更容易推导。 </p> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=4" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>张量可以表述为一个值的序列，用一个向量值的定义域和一个标量值的值域的函数表示。这些定义域中的向量是自然数的向量，而这些数字称为<a href="/wiki/%E6%8C%87%E6%A8%99" class="mw-disambig" title="指标">指标</a>。例如，取一3阶张量尺寸为2x5x7。这里，指标的范围从<1,1,1>到<2,5,7>。张量可以在指标为<1,1,1>有一个值，在指标为<1,1,2>有另一个值，等等一共70个值。（类似的，向量可以表示为一个值的序列，用一个标量值的定义域和一个标量值的值域的函数表示，定义域中的数字是自然数，称为指标，不同的指标的个数有时称为向量的<a href="/wiki/%E7%BB%B4%E5%BA%A6" class="mw-redirect" title="维度">维度</a>。） </p><p>一个张量<a href="/wiki/%E5%9C%BA" class="mw-disambig" title="场">场</a>是在<a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E7%A9%BA%E9%97%B4" title="欧几里得空间">欧几里得空间</a>中的每一点都给定一个张量值。这样不是像上面的例子中简单的有70个值，对于一个3阶张量，维度为<2,5,7>，空间中的每一个点有70个值和它相关。换句话说，张量场表示某个张量值的函数，其定义域为欧几里得空间。不是所有的函数都行—更多关于这些要求的细节参看<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a>。 </p><p>不是所有自然中的关系都是线性的，但是很多是<a href="/wiki/%E5%8F%AF%E5%BE%AE" class="mw-redirect" title="可微">可微</a>的因而可以局部的用<a href="/wiki/%E5%A4%9A%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" class="mw-redirect" title="多线性映射">多线性映射</a>来局部的逼近。这样多数物理学中的量都可以用张量表示。 </p><p>作为一个简单的例子，考虑水中的船。我们要描述它对受力的反应。力是一个向量，而船的反应是一个加速度，它也是一个向量。通常加速度不是和受力的方向相同，因为船体的特定形状。但是，这个力和加速之间的关系实际上是<a href="/wiki/%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90" class="mw-redirect" title="线性算子">线性</a>的。这样一个关系可以用一个（1,1）类型（也就是说，它把一个向量变成另一个向量）的张量表示。这个张量可以用<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>表示，当它乘以一个向量时就得到另一个作为结果。坐标系改变的时候，表示一个向量的数字会改变，同样，表示这个张量的矩阵中的数字也会改变。 </p><p>工程上，<a href="/wiki/%E5%88%9A%E4%BD%93" title="刚体">刚体</a>或<a href="/wiki/%E6%B5%81%E4%BD%93" title="流体">流体</a>内的<a href="/wiki/%E6%87%89%E5%8A%9B" title="应力">应力</a>也用一个张量表示；"张量"一词的拉丁语就表示引起张力的某种拉伸。如果材料内的一个特定的表面元素被选出来，在表面一侧的材料会对另一侧的施加一个力。通常，该力不和表面正交，但是它将线性的依赖于表面的朝向。这可以精确用（2,0）类型的张量精确的描述，或者更精确地说，是用一个类型为（2,0）的张量<i>场</i>来表示，因为张量可能在每一个不同。 </p><p>另外一些著名的几何中张量的例子有<a href="/wiki/%E4%BA%8C%E6%AC%A1%E5%9E%8B" title="二次型">二次型</a>，以及<a href="/wiki/%E6%9B%B2%E7%8E%87%E5%BC%A0%E9%87%8F" class="mw-disambig" title="曲率张量">曲率张量</a>。物理张量的例子有<a href="/wiki/%E8%83%BD%E5%8A%A8%E5%BC%A0%E9%87%8F" class="mw-redirect" title="能动张量">能动张量</a>，<a href="/wiki/%E6%83%AF%E9%87%8F" class="mw-redirect" title="惯量">惯量</a>和<a href="/w/index.php?title=%E6%9E%81%E5%8C%96%E5%BC%A0%E9%87%8F&action=edit&redlink=1" class="new" title="极化张量（页面不存在）">极化张量</a>。 </p><p>几何和物理的量可以通过考虑它们的表述内在的<a href="/wiki/%E8%87%AA%E7%94%B1%E5%BA%A6" class="mw-disambig" title="自由度">自由度</a>来分类。标量是那些可以用一个数表示的 --- <a href="/wiki/%E9%80%9F%E7%8E%87" title="速率">速率</a>，<a href="/wiki/%E8%B4%A8%E9%87%8F" title="质量">质量</a>，<a href="/wiki/%E6%B8%A9%E5%BA%A6" title="温度">温度</a>，等等。有一些向量类型的量，例如<a href="/wiki/%E5%8A%9B" title="力">力</a>，它需要一个数字的列表来表述。最后，像二次型这样的量需要用多维数组来表示。后面这些量只能视为张量。 </p><p>实际上，张量的概念相当广泛，可以用于上面所有的例子；标量和向量是张量的特殊情况。区别标量和向量以及区别这两者和更一般的张量的特征是表示它们的数组的指标的个数。这个个数称为张量的<b>阶</b>。这样，标量是0阶张量（不需要任何指标），而向量是一阶张量。 </p><p>张量的另外一个例子是<a href="/wiki/%E5%B9%BF%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA" class="mw-redirect" title="广义相对论">广义相对论</a>中的<a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E7%8E%87%E5%BC%A0%E9%87%8F" class="mw-redirect" title="黎曼曲率张量">黎曼曲率张量</a>，它是维度为<4,4,4,4>（3个空间维度 +时间维度 = 4个维度）的4阶张量。它可以当作256个分量（256 = 4×4×4×4）的矩阵（或者向量，其实是个4维数组）。只有20个分量是互相独立的，这个事实可以大大简化它的实际表达。 </p> <div class="mw-heading mw-heading2"><h2 id="方法细节"><span id=".E6.96.B9.E6.B3.95.E7.BB.86.E8.8A.82"></span>方法细节</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=5" title="编辑章节：方法细节"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>有几种想象和操作张量的<b>等价</b>方法；只有熟悉这个课题，才能了解其内容是等价的。 </p> <ul><li><b><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%BB%8F%E5%85%B8%E5%A4%84%E7%90%86&action=edit&redlink=1" class="new" title="张量经典处理（页面不存在）">经典方法</a></b></li></ul> <dl><dd>经典的方法把张量视为多维<a href="/wiki/%E6%95%B0%E7%BB%84" title="数组">数组</a>，它们是标量，1维向量和2维<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>的<i>n</i>维推广。张量的"分量"是数组中的值。这个思想可以进一步推广到<a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a>，那里张量的元素是<a href="/wiki/%E5%87%BD%E6%95%B0" title="函数">函数</a>，甚至<a href="/wiki/%E5%BE%AE%E5%88%86" title="微分">微分</a>。</dd></dl> <dl><dd>张量场理论在这个方法中大致可以视为<a href="/wiki/%E9%9B%85%E5%8F%AF%E6%AF%94%E7%9F%A9%E9%98%B5" title="雅可比矩阵">雅可比矩阵</a>的思想的推广。</dd></dl> <ul><li><b><a href="/wiki/%E5%BC%A0%E9%87%8F_(%E5%86%85%E8%95%B4%E5%AE%9A%E4%B9%89)" title="张量 (内蕴定义)">现代方法</a></b></li></ul> <dl><dd>现代（无分量）方法把张量首先视为抽象对象，表达了多线性概念的某种确定类型。其著名的性质可以从其定义导出，作为线性映射或者更一般的情况；而操作张量的规则作为从<a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>到<a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="多重线性代数">多重线性代数</a>的推广出现。这个处理方法在高等的研究中大量的取代了基于分量的方法，其方式是更现代的无分量向量方法在基于分量的方法用于给出向量概念的基本引例之后就取代传统的基于分量的方法。可以说，口号就是“张量是某个张量空间的元素”。</dd></dl> <ul><li><b><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%9A%84%E4%B8%AD%E9%97%B4%E5%A4%84%E7%90%86&action=edit&redlink=1" class="new" title="张量的中间处理（页面不存在）">张量的中间处理</a></b>条目试图为两个极端建立联系，并显示他们的关系。</li></ul> <p>最终，同样的计算内容被表达出来，两种方式都可以。技术性术语列表请参看<a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%90%86%E8%AE%BA%E8%AF%8D%E6%B1%87&action=edit&redlink=1" class="new" title="张量理论词汇（页面不存在）">张量理论词汇</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="张量密度"><span id=".E5.BC.A0.E9.87.8F.E5.AF.86.E5.BA.A6"></span>张量密度</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=6" title="编辑章节：张量密度"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a>也可有一个“密度”。密度为<i>r</i>的张量和普通张量一样坐标变换，但是它还要乘以<a href="/wiki/%E9%9B%85%E5%8F%AF%E6%AF%94%E7%9F%A9%E9%98%B5" title="雅可比矩阵">雅可比矩阵</a>的行列式值的第<i>r</i>次幂。这个的最佳解释可能是使用<a href="/wiki/%E5%90%91%E9%87%8F%E4%B8%9B" title="向量丛">向量丛</a>：其中，<a href="/wiki/%E5%88%87%E4%B8%9B" title="切丛">切丛</a>的行列式丛是一个<a href="/wiki/%E7%BA%BF%E4%B8%9B" title="线丛">线丛</a>，可以用来'扭转'其它丛<i>r</i>次。 </p> <div class="mw-heading mw-heading2"><h2 id="張量阶"><span id=".E5.BC.B5.E9.87.8F.E9.98.B6"></span>张量阶</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=7" title="编辑章节：张量阶"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">参见：<a href="/w/index.php?title=%E5%BC%B5%E9%87%8F%E6%A8%99%E6%BA%96%E6%A0%BC%E5%BC%8F&action=edit&redlink=1" class="new" title="张量标准格式（页面不存在）">张量标准格式</a></div> <table class="wikitable"> <tbody><tr valign="top"> <th align="left" bgcolor="#EEEEEE">等级 </th> <th align="left" bgcolor="#EEEEEE">别名 </th> <th align="left" bgcolor="#EEEEEE">记号 </th> <th align="left" bgcolor="#EEEEEE">一般变换 </th> <th align="left" bgcolor="#EEEEEE">张量密度变换方式<sup>*</sup> </th></tr> <tr> <td align="center">0</td> <td align="left"><a href="/wiki/%E6%A0%87%E9%87%8F" class="mw-disambig" title="标量">标量</a> </td> <td align="center">S </td> <td align="left">S'=S</td> <td align="left">S'=|a|S </td></tr> <tr> <td align="center">1</td> <td align="left"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" class="mw-redirect" title="向量空间">（馀）向量</a> </td> <td align="center">V<sub>i</sub> </td> <td align="left">V'<sub>i</sub>=a<sub>i</sub><sup>j</sup>V<sub>j</sub> </td> <td align="left">V'<sub>i</sub>=|a|a<sub>i</sub><sup>j</sup>V<sub>j</sub> </td></tr> <tr> <td align="center">2</td> <td align="left"><a href="/wiki/%E7%9F%A9%E9%99%A3" class="mw-redirect" title="矩阵">（共变）矩阵</a> </td> <td align="center">M<sub>i</sub><sub>j</sub> </td> <td align="left">M'<sub>ij</sub>=a<sub>i</sub><sup>k</sup>a<sub>j</sub><sup>l</sup>M<sub>kl</sub> </td> <td align="left">M'<sub>ij</sub>=|a|a<sub>i</sub><sup>k</sup>a<sub>j</sub><sup>l</sup>M<sub>kl</sub> </td></tr> <tr> <td align="center">3</td> <td align="left">（共变）3阶张量 </td> <td align="center">T<sub>ijk</sub> </td> <td align="left">T'<sub>ijk</sub>=a<sub>i</sub><sup>l</sup>a<sub>j</sub><sup>s</sup>a<sub>k</sub><sup>m</sup>T<sub>lsm</sub> </td> <td align="left">T'<sub>ijk</sub>=|a|a<sub>i</sub><sup>l</sup>a<sub>j</sub><sup>s</sup>a<sub>k</sub><sup>m</sup>T<sub>lsm</sub> </td></tr></tbody></table> <p>其中，a<sub>i</sub><sup>j</sup>是坐标变换的雅可比矩阵。这里所有的分量假定为共变，反变的张量变换要用a的逆矩阵。注意这里是用<a href="/wiki/%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6%E8%AE%B0%E5%8F%B7" class="mw-redirect" title="爱因斯坦记号">爱因斯坦记号</a>。 </p><p><i><sup>*</sup> |a|是a<sub>i</sub><sup>j</sup>的行列式。</i> </p> <div class="mw-heading mw-heading2"><h2 id="参閱"><span id=".E5.8F.82.E9.96.B1"></span>参阅</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=8" title="编辑章节：参阅"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%90%86%E8%AE%BA%E8%AF%8D%E6%B1%87&action=edit&redlink=1" class="new" title="张量理论词汇（页面不存在）">张量理论词汇</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="记法常规"><span id=".E8.AE.B0.E6.B3.95.E5.B8.B8.E8.A7.84"></span>记法常规</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=9" title="编辑章节：记法常规"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6%E8%AE%B0%E5%8F%B7" class="mw-redirect" title="爱因斯坦记号">爱因斯坦记号</a></li> <li><a href="/w/index.php?title=Voigt%E8%AE%B0%E5%8F%B7&action=edit&redlink=1" class="new" title="Voigt记号（页面不存在）">Voigt记号</a>（福格特）</li> <li><a href="/wiki/%E6%8A%BD%E8%B1%A1%E6%8C%87%E6%A0%87%E8%AE%B0%E5%8F%B7" title="抽象指标记号">抽象指标记号</a></li> <li><a href="/wiki/%E5%BA%A6%E9%87%8F%E5%BC%A0%E9%87%8F" title="度量张量">度量张量</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="基础"><span id=".E5.9F.BA.E7.A1.80"></span>基础</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=10" title="编辑章节：基础"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%85%B1%E5%8F%98" class="mw-disambig" title="共变">共变</a>（协变）</li> <li><a href="/wiki/%E5%8F%8D%E5%8F%98" class="mw-disambig" title="反变">反变</a>（逆变）</li> <li><a href="/wiki/1-%E5%BD%A2%E5%BC%8F" title="1-形式">1-形式</a></li> <li><a href="/wiki/%E5%BC%A0%E9%87%8F%E7%A7%AF" title="张量积">张量积</a></li> <li><a href="/wiki/%E7%BA%A4%E7%BB%B4%E4%B8%9B" title="纤维丛">纤维丛</a></li> <li><a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="应用"><span id=".E5.BA.94.E7.94.A8"></span>应用</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=11" title="编辑章节：应用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E5%AF%BC%E6%95%B0&action=edit&redlink=1" class="new" title="张量导数（页面不存在）">张量导数</a></li> <li><a href="/wiki/%E7%BB%9D%E5%AF%B9%E5%BE%AE%E5%88%86" class="mw-redirect" title="绝对微分">绝对微分</a></li> <li><a href="/wiki/%E6%9B%B2%E7%8E%87" title="曲率">曲率</a></li> <li><a href="/wiki/%E9%BB%8E%E6%9B%BC%E5%87%A0%E4%BD%95" title="黎曼几何">黎曼几何</a></li> <li><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%90%86%E8%AE%BA%E5%9C%A8%E5%B7%A5%E7%A8%8B%E7%A7%91%E5%AD%A6%E4%B8%AD%E7%9A%84%E5%BA%94%E7%94%A8&action=edit&redlink=1" class="new" title="张量理论在工程科学中的应用（页面不存在）">张量理论在工程科学中的应用</a></li> <li><a href="/w/index.php?title=%E5%BC%A0%E9%87%8F%E7%90%86%E8%AE%BA%E5%9C%A8%E7%89%A9%E7%90%86%E4%B8%AD%E7%9A%84%E5%BA%94%E7%94%A8&action=edit&redlink=1" class="new" title="张量理论在物理中的应用（页面不存在）">张量理论在物理中的应用</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考资料"><span id=".E5.8F.82.E8.80.83.E8.B5.84.E6.96.99"></span>参考资料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=12" title="编辑章节：参考资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation book">Lee, J.M. <a rel="nofollow" class="external text" href="https://archive.org/details/riemannianmanifo00leej_627">Riemannian Manifolds</a>. Springer. 1997: <a rel="nofollow" class="external text" href="https://archive.org/details/riemannianmanifo00leej_627/page/n28">12</a>. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0387983228" title="Special:网络书源/0387983228"><span title="国际标准书号">ISBN</span> 0387983228</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%BC%B5%E9%87%8F&rft.aufirst=J.M.&rft.aulast=Lee&rft.btitle=Riemannian+Manifolds&rft.date=1997&rft.genre=book&rft.isbn=0387983228&rft.pages=12&rft.pub=Springer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Friemannianmanifo00leej_627&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="參考書籍"><span id=".E5.8F.83.E8.80.83.E6.9B.B8.E7.B1.8D"></span>参考书籍</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=13" title="编辑章节：参考书籍"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Tensors, Differential Forms, and Variational Principles (1989) David Lovelock, Hanno Rund</li> <li>Tensor Analysis on Manifolds (1981) Richard L Bishop, Samuel I. Goldberg</li> <li>Introduction to Tensor Calculus, Relativity and Cosmology (2003) D. F. Lawden</li> <li>Tensor Analysis (2003) L.P. Lebedev, Michael J. Cloud</li> <li>Calculus of Variations (2000) S. V. Fomin, I. M. Gelfand</li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=14" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051104201543/http://nrich.maths.org/askedNRICH/edited/2604.html">各种教授张量的方法的讨论，以及教科书的推荐</a></li> <li><a rel="nofollow" class="external text" href="https://www.physicsforums.com/showthread.php?t=35920&page=1&pp=15">讨论基本和细节的定义以及例子的话题</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20051216124720/http://www.physicsforums.com/showthread.php?t=35920&page=1&pp=15">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200106103811/http://www.math.odu.edu/~jhh/counter2.html">张量微积分和连续力学简介</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="张量软件"><span id=".E5.BC.A0.E9.87.8F.E8.BD.AF.E4.BB.B6"></span>张量软件</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F&action=edit&section=15" title="编辑章节：张量软件"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://grtensor.phy.queensu.ca/">GRTensorII</a> <a href="/wiki/%E7%BE%8E%E5%9C%8B%E5%9C%8B%E6%9C%83%E5%9C%96%E6%9B%B8%E9%A4%A8" class="mw-redirect" title="美国国会图书馆">美国国会图书馆</a>的<a rel="nofollow" class="external text" href="http://webarchive.loc.gov/all/20020914105414/http%3A//grtensor.phy.queensu.ca/">存档</a>，存档日期2002-09-14执行微分几何一般领域中的计算的计算机代数包。GRTensor II不是独立的软件包，该程序通过Maple 9.5和所有Maple V第3版的版本一起运行。一个受限版（GRTensorM）已经移植到Mathematica上。</li> <li><a rel="nofollow" class="external text" href="http://maxima.sourceforge.net/">maxima</a>是一个<a href="/wiki/GPL" class="mw-redirect" title="GPL">GPL</a> <a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E4%BB%A3%E6%95%B0%E7%B3%BB%E7%BB%9F" class="mw-redirect" title="计算机代数系统">计算机代数系统</a><a href="/wiki/%E8%87%AA%E7%94%B1%E8%BD%AF%E4%BB%B6" title="自由软件">自由软件</a>，它可以用来做张量代数计算。 <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050214092040/http://maxima.sourceforge.net/docs/original/maxima_28.html#SEC89">maxima中的张量</a></li></ul></li> <li><a rel="nofollow" class="external text" href="http://www.math.washington.edu/~lee/Ricci/">Ricci</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20051025161358/http://www.math.washington.edu/~lee/Ricci/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）是用于Mathematica 2.x的一个系统，后来也用于基本的张量分析，可免费得到。</li> <li><a rel="nofollow" class="external autonumber" href="https://cadabra.science/">[1]</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20190723110951/https://cadabra.science/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） <a rel="nofollow" class="external autonumber" href="https://github.com/kpeeters/cadabra2">[2]</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200610221908/https://github.com/kpeeters/cadabra2">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）Cadabra 是一款为研究张量分析和场论而设计的计算机代数软件</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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.navbox-title{background-color:#ccf;position:relative}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em;width:auto;padding-left:0.2em;position:absolute;left:1em}.mw-parser-output .navbox 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title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E5%BC%B5%E9%87%8F" title="Template talk:张量"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E5%BC%B5%E9%87%8F" title="Special:编辑页面/Template:张量"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="張量" style="font-size:110%;margin:0 5em"><a class="mw-selflink selflink">张量</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span class="ilh-all" data-orig-title="張量理論中的字彙" data-lang-code="en" data-lang-name="英语" data-foreign-title="Glossary of tensor theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F%E7%90%86%E8%AB%96%E4%B8%AD%E7%9A%84%E5%AD%97%E5%BD%99&action=edit&redlink=1" class="new" title="张量理论中的字汇（页面不存在）">张量理论中的字汇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Glossary_of_tensor_theory" class="extiw" title="en:Glossary of tensor theory"><span lang="en" dir="auto">Glossary of tensor theory</span></a></span>）</span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">范畴</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9D%90%E6%A8%99%E7%B3%BB" title="坐标系">坐标系</a></li> <li><a href="/wiki/%E5%A4%9A%E9%87%8D%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="多重线性代数">多重线性代数</a></li> <li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">欧几里得几何</a></li> <li><a href="/wiki/%E5%BC%A0%E9%87%8F%E4%BB%A3%E6%95%B0" title="张量代数">张量代数</a></li> <li><a href="/wiki/%E4%B8%A6%E7%9F%A2%E5%BC%B5%E9%87%8F" title="并矢张量">并矢张量</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="微分几何">微分几何</a></li> <li><a href="/wiki/%E5%A4%96%E5%BE%AE%E5%88%86" title="外微分">外微分</a></li> <li><a class="mw-selflink selflink">张量微积分</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6" title="物理学">物理学</a></li><li><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%A6" title="工程学">工程学</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E4%BB%8B%E8%B4%A8%E5%8A%9B%E5%AD%A6" title="连续介质力学">连续介质力学</a></li> <li><a href="/wiki/%E7%94%B5%E7%A3%81%E5%AD%A6" title="电磁学">电磁学</a></li> <li><a href="/wiki/%E8%BE%93%E8%BF%90%E7%8E%B0%E8%B1%A1" title="输运现象">输运现象</a></li> <li><a href="/wiki/%E5%BB%A3%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96" title="广义相对论">广义相对论</a></li> <li><a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E8%A7%86%E8%A7%89" title="计算机视觉">计算机视觉</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">符号</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="指標記號" data-lang-code="en" data-lang-name="英语" data-foreign-title="index notation"><span class="ilh-page"><a href="/w/index.php?title=%E6%8C%87%E6%A8%99%E8%A8%98%E8%99%9F&action=edit&redlink=1" class="new" title="指标记号（页面不存在）">指标记号</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/index_notation" class="extiw" title="en:index notation"><span lang="en" dir="auto">index notation</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%A4%9A%E9%87%8D%E6%8C%87%E6%A0%87" title="多重指标">多重指标</a></li> <li><a href="/wiki/%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6%E6%B1%82%E5%92%8C%E7%BA%A6%E5%AE%9A" title="爱因斯坦求和约定">爱因斯坦求和约定</a></li> <li><span class="ilh-all" data-orig-title="瑞奇微積分" data-lang-code="en" data-lang-name="英语" data-foreign-title="Ricci calculus"><span class="ilh-page"><a href="/w/index.php?title=%E7%91%9E%E5%A5%87%E5%BE%AE%E7%A9%8D%E5%88%86&action=edit&redlink=1" class="new" title="瑞奇微积分（页面不存在）">瑞奇微积分</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Ricci_calculus" class="extiw" title="en:Ricci calculus"><span lang="en" dir="auto">Ricci calculus</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%BD%98%E6%B4%9B%E6%96%AF%E5%9C%96%E5%BD%A2%E7%AC%A6%E8%99%9F" title="潘洛斯图形符号">潘洛斯图形符号</a></li> <li><span class="ilh-all" data-orig-title="佛依格特記號" data-lang-code="en" data-lang-name="英语" data-foreign-title="Voigt notation"><span class="ilh-page"><a href="/w/index.php?title=%E4%BD%9B%E4%BE%9D%E6%A0%BC%E7%89%B9%E8%A8%98%E8%99%9F&action=edit&redlink=1" class="new" title="佛依格特记号（页面不存在）">佛依格特记号</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Voigt_notation" class="extiw" title="en:Voigt notation"><span lang="en" dir="auto">Voigt notation</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%8A%BD%E8%B1%A1%E6%8C%87%E6%A0%87%E8%AE%B0%E5%8F%B7" title="抽象指标记号">抽象指标记号</a></li> <li><span class="ilh-all" data-orig-title="四元組 (指標記號)" data-lang-code="en" data-lang-name="英语" data-foreign-title="tetrad (index notation)"><span class="ilh-page"><a href="/w/index.php?title=%E5%9B%9B%E5%85%83%E7%B5%84_(%E6%8C%87%E6%A8%99%E8%A8%98%E8%99%9F)&action=edit&redlink=1" class="new" title="四元组 (指标记号)（页面不存在）">四元组 (指标记号)</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/tetrad_(index_notation)" class="extiw" title="en:tetrad (index notation)"><span lang="en" dir="auto">tetrad (index notation)</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="凡得瓦登記號" data-lang-code="en" data-lang-name="英语" data-foreign-title="Van der Waerden notation"><span class="ilh-page"><a href="/w/index.php?title=%E5%87%A1%E5%BE%97%E7%93%A6%E7%99%BB%E8%A8%98%E8%99%9F&action=edit&redlink=1" class="new" title="凡得瓦登记号（页面不存在）">凡得瓦登记号</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Van_der_Waerden_notation" class="extiw" title="en:Van der Waerden notation"><span lang="en" dir="auto">Van der Waerden notation</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">张量定义</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BC%A0%E9%87%8F_(%E5%86%85%E8%95%B4%E5%AE%9A%E4%B9%89)" title="张量 (内蕴定义)">张量 (内蕴定义)</a></li> <li><a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a></li> <li><span class="ilh-all" data-orig-title="張量密度" data-lang-code="en" data-lang-name="英语" data-foreign-title="tensor density"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F%E5%AF%86%E5%BA%A6&action=edit&redlink=1" class="new" title="张量密度（页面不存在）">张量密度</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/tensor_density" class="extiw" title="en:tensor density"><span lang="en" dir="auto">tensor density</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="曲線座標中的張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="tensors in curvilinear coordinates"><span class="ilh-page"><a href="/w/index.php?title=%E6%9B%B2%E7%B7%9A%E5%BA%A7%E6%A8%99%E4%B8%AD%E7%9A%84%E5%BC%B5%E9%87%8F&action=edit&redlink=1" class="new" title="曲线座标中的张量（页面不存在）">曲线座标中的张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/tensors_in_curvilinear_coordinates" class="extiw" title="en:tensors in curvilinear coordinates"><span lang="en" dir="auto">tensors in curvilinear coordinates</span></a></span>）</span></span></li> <li><a class="mw-selflink selflink">混合张量</a></li> <li><span class="ilh-all" data-orig-title="反對稱張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="antisymmetric tensor"><span class="ilh-page"><a href="/wiki/%E5%8F%8D%E5%AF%B9%E7%A7%B0%E5%BC%A0%E9%87%8F" title="反对称张量">反对称张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/antisymmetric_tensor" class="extiw" title="en:antisymmetric tensor"><span lang="en" dir="auto">antisymmetric tensor</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="對稱張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="symmetric tensor"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E5%BC%B5%E9%87%8F&action=edit&redlink=1" class="new" title="对称张量（页面不存在）">对称张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/symmetric_tensor" class="extiw" title="en:symmetric tensor"><span lang="en" dir="auto">symmetric tensor</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="張量算符" data-lang-code="en" data-lang-name="英语" data-foreign-title="tensor operator"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F%E7%AE%97%E7%AC%A6&action=edit&redlink=1" class="new" title="张量算符（页面不存在）">张量算符</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/tensor_operator" class="extiw" title="en:tensor operator"><span lang="en" dir="auto">tensor operator</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%BC%A0%E9%87%8F%E5%9C%BA" title="张量场">张量场</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%BF%90%E7%AE%97" title="运算">运算</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BC%A0%E9%87%8F%E7%A7%AF" title="张量积">张量积</a></li> <li><a href="/wiki/%E5%A4%96%E4%BB%A3%E6%95%B0" title="外代数">外代数</a></li> <li><span class="ilh-all" data-orig-title="張量收縮" data-lang-code="en" data-lang-name="英语" data-foreign-title="tensor contraction"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B5%E9%87%8F%E6%94%B6%E7%B8%AE&action=edit&redlink=1" class="new" title="张量收缩（页面不存在）">张量收缩</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/tensor_contraction" class="extiw" title="en:tensor contraction"><span lang="en" dir="auto">tensor contraction</span></a></span>）</span></span></li> <li><a href="/wiki/%E8%BD%AC%E7%BD%AE%E7%9F%A9%E9%98%B5" title="转置矩阵">转置矩阵</a></li> <li><a href="/wiki/%E6%8C%87%E6%A0%87%E7%9A%84%E4%B8%8A%E5%8D%87%E5%92%8C%E4%B8%8B%E9%99%8D" title="指标的上升和下降">指标的上升和下降</a></li> <li><a href="/wiki/%E9%9C%8D%E5%A5%87%E5%AF%B9%E5%81%B6" title="霍奇对偶">霍奇对偶</a></li> <li><a href="/wiki/%E5%8D%8F%E5%8F%98%E5%BE%AE%E5%95%86" title="协变微商">协变微商</a></li> <li><a href="/wiki/%E5%A4%96%E5%BE%AE%E5%88%86" title="外微分">外微分</a></li> <li><a href="/wiki/%E5%A4%96%E5%85%B1%E5%8F%98%E5%AF%BC%E6%95%B0" title="外共变导数">外共变导数</a></li> <li><a href="/wiki/%E6%9D%8E%E5%AF%BC%E6%95%B0" title="李导数">李导数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">相关抽象名词</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%B6%AD%E5%BA%A6" title="维度">维度</a></li> <li><a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (线性代数)">基 (线性代数)</a></li> <li><span class="ilh-all" data-orig-title="向量 (數學和物理)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Vector (mathematics and physics)"><span class="ilh-page"><a href="/w/index.php?title=%E5%90%91%E9%87%8F_(%E6%95%B8%E5%AD%B8%E5%92%8C%E7%89%A9%E7%90%86)&action=edit&redlink=1" class="new" title="向量 (数学和物理)（页面不存在）">向量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)" class="extiw" title="en:Vector (mathematics and physics)"><span lang="en" dir="auto">Vector (mathematics and physics)</span></a></span>）</span></span>、<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a></li> <li><span class="ilh-all" data-orig-title="多重向量" data-lang-code="en" data-lang-name="英语" data-foreign-title="multivector"><span class="ilh-page"><a href="/w/index.php?title=%E5%A4%9A%E9%87%8D%E5%90%91%E9%87%8F&action=edit&redlink=1" class="new" title="多重向量（页面不存在）">多重向量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/multivector" class="extiw" title="en:multivector"><span lang="en" dir="auto">multivector</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%85%B1%E8%AE%8A%E5%92%8C%E5%8F%8D%E8%AE%8A" title="共变和反变">共变和反变</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性映射</a></li> <li><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></li> <li><a href="/wiki/%E6%97%8B%E9%87%8F" title="旋量">旋量</a></li> <li><span class="ilh-all" data-orig-title="卡當形式 (物理學)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Cartan formalism (physics)"><span class="ilh-page"><a href="/w/index.php?title=%E5%8D%A1%E7%95%B6%E5%BD%A2%E5%BC%8F_(%E7%89%A9%E7%90%86%E5%AD%B8)&action=edit&redlink=1" class="new" title="卡当形式 (物理学)（页面不存在）">卡当形式 (物理学)</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Cartan_formalism_(physics)" class="extiw" title="en:Cartan formalism (physics)"><span lang="en" dir="auto">Cartan formalism (physics)</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%BD%A2%E5%BC%8F" title="微分形式">微分形式</a></li> <li><a href="/wiki/%E5%BE%AE%E5%88%86%E5%BD%A2%E5%BC%8F" title="微分形式">微分形式</a></li> <li><a href="/wiki/%E8%81%94%E7%BB%9C%E5%BD%A2%E5%BC%8F" title="联络形式">联络形式</a></li> <li><a href="/wiki/%E6%B5%8B%E5%9C%B0%E7%BA%BF" title="测地线">测地线</a></li> <li><a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a></li> <li><a href="/wiki/%E7%BA%A4%E7%BB%B4%E4%B8%9B" title="纤维丛">纤维丛</a></li> <li><a href="/wiki/%E5%88%97%E7%BB%B4-%E5%A5%87%E7%BB%B4%E5%A1%94%E8%81%94%E7%BB%9C" title="列维-奇维塔联络">列维-奇维塔联络</a></li> <li><a href="/wiki/%E4%BB%BF%E5%B0%84%E8%81%94%E7%BB%9C" title="仿射联络">仿射联络</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">知名张量</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">数学</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%CE%B4%E5%87%BD%E6%95%B0" title="克罗内克δ函数">克罗内克δ函数</a></li> <li><a href="/wiki/%E5%88%97%E7%B6%AD-%E5%A5%87%E7%B6%AD%E5%A1%94%E7%AC%A6%E8%99%9F" title="列维-奇维塔符号">列维-奇维塔符号</a></li> <li><a href="/wiki/%E5%BA%A6%E9%87%8F%E5%BC%A0%E9%87%8F" title="度量张量">度量张量</a></li> <li><span class="ilh-all" data-orig-title="nonmetricity tensor" data-lang-code="en" data-lang-name="英语" data-foreign-title="nonmetricity tensor"><span class="ilh-page"><a href="/w/index.php?title=Nonmetricity_tensor&action=edit&redlink=1" class="new" title="Nonmetricity tensor（页面不存在）">nonmetricity tensor</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/nonmetricity_tensor" class="extiw" title="en:nonmetricity tensor"><span lang="en" dir="auto">nonmetricity tensor</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%85%8B%E9%87%8C%E6%96%AF%E6%89%98%E8%B4%B9%E5%B0%94%E7%AC%A6%E5%8F%B7" title="克里斯托费尔符号">克里斯托费尔符号</a></li> <li><a href="/wiki/%E9%87%8C%E5%A5%87%E6%9B%B2%E7%8E%87%E5%BC%B5%E9%87%8F" title="里奇曲率张量">里奇曲率张量</a></li> <li><a href="/wiki/%E9%BB%8E%E6%9B%BC%E6%9B%B2%E7%8E%87%E5%BC%B5%E9%87%8F" title="黎曼曲率张量">黎曼曲率张量</a></li> <li><span class="ilh-all" data-orig-title="外勒張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="Weyl tensor"><span class="ilh-page"><a href="/w/index.php?title=%E5%A4%96%E5%8B%92%E5%BC%B5%E9%87%8F&action=edit&redlink=1" class="new" title="外勒张量（页面不存在）">外勒张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Weyl_tensor" class="extiw" title="en:Weyl tensor"><span lang="en" dir="auto">Weyl tensor</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%89%AD%E7%8E%87%E5%BC%B5%E9%87%8F" title="扭率张量">扭率张量</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">物理</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F" title="转动惯量">转动惯量</a></li> <li><a href="/wiki/%E8%A7%92%E5%8A%A8%E9%87%8F" title="角动量">角动量</a></li> <li><span class="ilh-all" data-orig-title="自旋張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="spin tensor"><span class="ilh-page"><a href="/w/index.php?title=%E8%87%AA%E6%97%8B%E5%BC%B5%E9%87%8F&action=edit&redlink=1" class="new" title="自旋张量（页面不存在）">自旋张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/spin_tensor" class="extiw" title="en:spin tensor"><span lang="en" dir="auto">spin tensor</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%9F%AF%E8%A5%BF%E5%BA%94%E5%8A%9B%E5%BC%A0%E9%87%8F" title="柯西应力张量">柯西应力张量</a></li> <li><a href="/wiki/%E6%87%89%E5%8A%9B-%E8%83%BD%E9%87%8F%E5%BC%B5%E9%87%8F" title="应力-能量张量">应力-能量张量</a></li> <li><a href="/wiki/%E9%9B%BB%E7%A3%81%E5%BC%B5%E9%87%8F" title="电磁张量">电磁张量</a></li> <li><span class="ilh-all" data-orig-title="膠子場強度張量" data-lang-code="en" data-lang-name="英语" data-foreign-title="gluon field strength tensor"><span class="ilh-page"><a href="/w/index.php?title=%E8%86%A0%E5%AD%90%E5%A0%B4%E5%BC%B7%E5%BA%A6%E5%BC%B5%E9%87%8F&action=edit&redlink=1" class="new" title="胶子场强度张量（页面不存在）">胶子场强度张量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/gluon_field_strength_tensor" class="extiw" title="en:gluon field strength tensor"><span lang="en" dir="auto">gluon field strength tensor</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6%E5%BC%A0%E9%87%8F" title="爱因斯坦张量">爱因斯坦张量</a></li> <li><span class="ilh-all" data-orig-title="度量張量 (廣義相對論)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Metric tensor (general relativity)"><span class="ilh-page"><a href="/wiki/%E5%BA%A6%E9%87%8F%E5%BC%A0%E9%87%8F_(%E5%B9%BF%E4%B9%89%E7%9B%B8%E5%AF%B9%E8%AE%BA)" class="mw-redirect" title="度量张量 (广义相对论)">度量张量 (广相)</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)" class="extiw" title="en:Metric tensor (general relativity)"><span lang="en" dir="auto">Metric tensor (general relativity)</span></a></span>）</span></span></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%AE%B6" title="数学家">数学家</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%90%8A%E6%98%82%E5%93%88%E5%BE%B7%C2%B7%E6%AD%90%E6%8B%89" title="莱昂哈德·欧拉">莱昂哈德·欧拉</a></li> <li><a href="/wiki/%E5%8D%A1%E7%88%BE%C2%B7%E5%BC%97%E9%87%8C%E5%BE%B7%E9%87%8C%E5%B8%8C%C2%B7%E9%AB%98%E6%96%AF" title="卡尔·弗里德里希·高斯">卡尔·弗里德里希·高斯</a></li> <li><a href="/wiki/%E5%A5%A7%E5%8F%A4%E6%96%AF%E4%B8%81-%E8%B7%AF%E6%98%93%C2%B7%E6%9F%AF%E8%A5%BF" title="奥古斯丁-路易·柯西">奥古斯丁-路易·柯西</a></li> <li><a href="/wiki/%E8%B5%AB%E7%88%BE%E6%9B%BC%C2%B7%E6%A0%BC%E6%8B%89%E6%96%AF%E6%9B%BC" title="赫尔曼·格拉斯曼">赫尔曼·格拉斯曼</a></li> <li><a href="/wiki/%E6%A0%BC%E9%9B%B7%E6%88%88%E9%87%8C%E5%A5%A5%C2%B7%E9%87%8C%E5%A5%87-%E5%BA%93%E5%B0%94%E5%B7%B4%E6%96%AF%E6%89%98%E7%BD%97" title="格雷戈里奥·里奇-库尔巴斯托罗">格雷戈里奥·里奇-库尔巴斯托罗</a></li> <li><a href="/wiki/%E5%9B%BE%E5%88%A9%E5%A5%A5%C2%B7%E5%88%97%E7%BB%B4-%E9%BD%90%E7%BB%B4%E5%A1%94" title="图利奥·列维-齐维塔">图利奥·列维-齐维塔</a></li> <li><span class="ilh-all" data-orig-title="Jan Arnoldus Schouten" data-lang-code="en" data-lang-name="英语" data-foreign-title="Jan Arnoldus Schouten"><span class="ilh-page"><a href="/w/index.php?title=Jan_Arnoldus_Schouten&action=edit&redlink=1" class="new" title="Jan Arnoldus Schouten（页面不存在）">Jan Arnoldus Schouten</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Jan_Arnoldus_Schouten" class="extiw" title="en:Jan Arnoldus Schouten"><span lang="en" dir="auto">Jan Arnoldus Schouten</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%B3%A2%E6%81%A9%E5%93%88%E5%BE%B7%C2%B7%E9%BB%8E%E6%9B%BC" class="mw-redirect" title="波恩哈德·黎曼">波恩哈德·黎曼</a></li> <li><a href="/wiki/%E5%9F%83%E5%B0%94%E6%B8%A9%C2%B7%E5%B8%83%E9%B2%81%E8%AF%BA%C2%B7%E5%85%8B%E9%87%8C%E6%96%AF%E6%89%98%E8%B4%B9%E5%B0%94" title="埃尔温·布鲁诺·克里斯托费尔">埃尔温·布鲁诺·克里斯托费尔</a></li> <li><span class="ilh-all" data-orig-title="沃爾德馬爾·佛依格特" data-lang-code="en" data-lang-name="英语" data-foreign-title="Woldemar Voigt"><span class="ilh-page"><a href="/w/index.php?title=%E6%B2%83%E7%88%BE%E5%BE%B7%E9%A6%AC%E7%88%BE%C2%B7%E4%BD%9B%E4%BE%9D%E6%A0%BC%E7%89%B9&action=edit&redlink=1" class="new" title="沃尔德马尔·佛依格特（页面不存在）">沃尔德马尔·佛依格特</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Woldemar_Voigt" class="extiw" title="en:Woldemar Voigt"><span lang="en" dir="auto">Woldemar Voigt</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%9F%83%E5%88%A9%C2%B7%E5%98%89%E5%BD%93" title="埃利·嘉当">埃利·嘉当</a></li> <li><a href="/wiki/%E8%B5%AB%E5%B0%94%E6%9B%BC%C2%B7%E5%A4%96%E5%B0%94" title="赫尔曼·外尔">赫尔曼·外尔</a></li> <li><a href="/wiki/%E9%98%BF%E5%B0%94%E4%BC%AF%E7%89%B9%C2%B7%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6" title="阿尔伯特·爱因斯坦">阿尔伯特·爱因斯坦</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84261037"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:%E8%A7%84%E8%8C%83%E6%8E%A7%E5%88%B6" title="Help:规范控制">规范控制数据库</a>：各地 <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a 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