物質微分 - Wikipedia

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vector-button-flush-right vector-user-menu-logged-out" title="その他の操作" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="個人用ツール" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">個人用ツール</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="利用者メニュー" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a 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mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> ログアウトした編集者のページ <a href="/wiki/Wikipedia:%E3%82%A6%E3%82%A3%E3%82%AD%E3%83%9A%E3%83%87%E3%82%A3%E3%82%A2%E3%81%B8%E3%82%88%E3%81%86%E3%81%93%E3%81%9D" aria-label="編集の詳細"><span>もっと詳しく</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/%E7%89%B9%E5%88%A5:%E8%87%AA%E5%88%86%E3%81%AE%E6%8A%95%E7%A8%BF%E8%A8%98%E9%8C%B2" title="このIPアドレスからなされた編集の一覧 [y]" accesskey="y"><span>投稿記録</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%88%E3%83%BC%E3%82%AF%E3%83%9A%E3%83%BC%E3%82%B8" title="このIPアドレスからなされた編集についての議論 [n]" accesskey="n"><span>トーク</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="サイト"> <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="目次" 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">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</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">ページ先頭</div> </a> </li> <li id="toc-定義" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>定義</span> </div> </a> <ul id="toc-定義-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-直観的意味" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#直観的意味"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>直観的意味</span> </div> </a> <ul id="toc-直観的意味-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-定常流" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定常流"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>定常流</span> </div> </a> <button aria-controls="toc-定常流-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>定常流サブセクションを切り替えます</span> </button> <ul id="toc-定常流-sublist" class="vector-toc-list"> <li id="toc-定常流における加速度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#定常流における加速度"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>定常流における加速度</span> </div> </a> <ul id="toc-定常流における加速度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ベルヌーイの定理と流線曲率の定理" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ベルヌーイの定理と流線曲率の定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>ベルヌーイの定理と流線曲率の定理</span> </div> </a> <ul id="toc-ベルヌーイの定理と流線曲率の定理-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-対流項" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#対流項"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>対流項</span> </div> </a> <button aria-controls="toc-対流項-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>対流項サブセクションを切り替えます</span> </button> <ul id="toc-対流項-sublist" class="vector-toc-list"> <li id="toc-曲線直交座標系" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#曲線直交座標系"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>曲線直交座標系</span> </div> </a> <ul id="toc-曲線直交座標系-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-相対論的物質微分" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相対論的物質微分"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>相対論的物質微分</span> </div> </a> <ul id="toc-相対論的物質微分-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-脚注" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#脚注"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>脚注</span> </div> </a> <button aria-controls="toc-脚注-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>脚注サブセクションを切り替えます</span> </button> <ul id="toc-脚注-sublist" class="vector-toc-list"> <li id="toc-出典" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#出典"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>出典</span> </div> </a> <ul id="toc-出典-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-sublist" class="vector-toc-list"> </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="目次" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="目次の表示・非表示を切り替え" > <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">目次の表示・非表示を切り替え</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"><span class="mw-page-title-main">物質微分</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <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="特定の記事の別の言語版に移動します。利用可能な言語16件" > <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-16" 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">16の言語版</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Materi%C3%A1lov%C3%A1_derivace" title="チェコ語: Materiálová derivace" lang="cs" hreflang="cs" data-title="Materiálová derivace" data-language-autonym="Čeština" data-language-local-name="チェコ語" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Materielt_afledte" title="デンマーク語: Materielt afledte" lang="da" hreflang="da" data-title="Materielt afledte" data-language-autonym="Dansk" data-language-local-name="デンマーク語" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Substantielle_Ableitung" title="ドイツ語: Substantielle Ableitung" lang="de" hreflang="de" data-title="Substantielle Ableitung" data-language-autonym="Deutsch" data-language-local-name="ドイツ語" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Material_derivative" title="英語: Material derivative" lang="en" hreflang="en" data-title="Material derivative" data-language-autonym="English" data-language-local-name="英語" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Derivada_material" title="スペイン語: Derivada material" lang="es" hreflang="es" data-title="Derivada material" data-language-autonym="Español" data-language-local-name="スペイン語" 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/Materiaalne_tuletis" title="エストニア語: Materiaalne tuletis" lang="et" hreflang="et" data-title="Materiaalne tuletis" data-language-autonym="Eesti" data-language-local-name="エストニア語" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B4%D8%AA%D9%82_%D9%85%D8%A7%D8%AF%D9%87" title="ペルシア語: مشتق ماده" lang="fa" hreflang="fa" data-title="مشتق ماده" data-language-autonym="فارسی" data-language-local-name="ペルシア語" 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/Materiaaliderivaatta" title="フィンランド語: Materiaaliderivaatta" lang="fi" hreflang="fi" data-title="Materiaaliderivaatta" data-language-autonym="Suomi" data-language-local-name="フィンランド語" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA_%D7%97%D7%95%D7%9E%D7%A8%D7%99%D7%AA" title="ヘブライ語: נגזרת חומרית" lang="he" hreflang="he" data-title="נגזרת חומרית" data-language-autonym="עברית" data-language-local-name="ヘブライ語" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Derivata_materiale" title="イタリア語: Derivata materiale" lang="it" hreflang="it" data-title="Derivata materiale" data-language-autonym="Italiano" data-language-local-name="イタリア語" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Materi%C3%ABle_afgeleide" title="オランダ語: Materiële afgeleide" lang="nl" hreflang="nl" data-title="Materiële afgeleide" data-language-autonym="Nederlands" data-language-local-name="オランダ語" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Operator_Stokesa" title="ポーランド語: Operator Stokesa" lang="pl" hreflang="pl" data-title="Operator Stokesa" data-language-autonym="Polski" data-language-local-name="ポーランド語" 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/Derivada_material" title="ポルトガル語: Derivada material" lang="pt" hreflang="pt" data-title="Derivada material" data-language-autonym="Português" data-language-local-name="ポルトガル語" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0%D1%8F_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0" title="ロシア語: Производная Лагранжа" lang="ru" hreflang="ru" data-title="Производная Лагранжа" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1o_h%C3%A0m_h%E1%BB%AFu_h%C3%ACnh" title="ベトナム語: Đạo hàm hữu hình" lang="vi" hreflang="vi" data-title="Đạo hàm hữu hình" data-language-autonym="Tiếng Việt" data-language-local-name="ベトナム語" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%89%A9%E8%B3%AA%E5%B0%8E%E6%95%B8" title="中国語: 物質導數" lang="zh" hreflang="zh" data-title="物質導數" data-language-autonym="中文" data-language-local-name="中国語" class="interlanguage-link-target"><span>中文</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/Q1801562#sitelinks-wikipedia" title="言語間リンクを編集" class="wbc-editpage">リンクを編集</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="名前空間"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86" title="本文を閲覧 [c]" accesskey="c"><span>ページ</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%E3%83%8E%E3%83%BC%E3%83%88:%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86" rel="discussion" title="本文ページについての議論 [t]" accesskey="t"><span>ノート</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="別の言語に切り替える" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">日本語</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="表示"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <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/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86"><span>閲覧</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=history" title="このページの過去の版 [h]" accesskey="h"><span>履歴表示</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="ページツール"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="ツール" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">ツール</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">ツール</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">非表示</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="その他の操作" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86"><span>閲覧</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=history"><span>履歴表示</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 全般 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%AA%E3%83%B3%E3%82%AF%E5%85%83/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86" title="ここにリンクしている全ウィキページの一覧 [j]" accesskey="j"><span>リンク元</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E9%96%A2%E9%80%A3%E3%83%9A%E3%83%BC%E3%82%B8%E3%81%AE%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%B3%81/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86" rel="nofollow" title="このページからリンクしているページの最近の更新 [k]" accesskey="k"><span>関連ページの更新状況</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB%E3%81%AE%E3%82%A2%E3%83%83%E3%83%97%E3%83%AD%E3%83%BC%E3%83%89" title="ファイルをアップロードする [u]" accesskey="u"><span>ファイルをアップロード</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E7%89%B9%E5%88%A5%E3%83%9A%E3%83%BC%E3%82%B8%E4%B8%80%E8%A6%A7" title="特別ページの一覧 [q]" accesskey="q"><span>特別ページ</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&oldid=91659822" title="このページのこの版への固定リンク"><span>この版への固定リンク</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=info" title="このページについての詳細情報"><span>ページ情報</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%81%93%E3%81%AE%E3%83%9A%E3%83%BC%E3%82%B8%E3%82%92%E5%BC%95%E7%94%A8&page=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&id=91659822&wpFormIdentifier=titleform" title="このページの引用方法"><span>このページを引用</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:UrlShortener&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E7%2589%25A9%25E8%25B3%25AA%25E5%25BE%25AE%25E5%2588%2586"><span>短縮URLを取得する</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:QrCode&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E7%2589%25A9%25E8%25B3%25AA%25E5%25BE%25AE%25E5%2588%2586"><span>QRコードをダウンロード</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> 印刷/書き出し </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%83%96%E3%83%83%E3%82%AF&bookcmd=book_creator&referer=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86"><span>ブックの新規作成</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:DownloadAsPdf&page=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=show-download-screen"><span>PDF 形式でダウンロード</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&printable=yes" title="このページの印刷用ページ [p]" accesskey="p"><span>印刷用バージョン</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 他のプロジェクト </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1801562" title="関連付けられたデータリポジトリ項目へのリンク [g]" accesskey="g"><span>ウィキデータ項目</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="ページツール"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="表示"> <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">表示</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">非表示</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> <div id="siteSub" class="noprint">出典: フリー百科事典『ウィキペディア（Wikipedia）』</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ja" dir="ltr"><table class="infobox" style="width: 200px; float: right; clear: right; text-align:center; font-size:90%; overflow:visible; transition: all 300ms 0s ease;"> <tbody><tr> <th style="position:sticky; top:0; background-color: #87cefa;"><a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6" title="連続体力学">連続体力学</a> </th></tr> <tr> <td><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:BernoullisLawDerivationDiagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/220px-BernoullisLawDerivationDiagram.svg.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/330px-BernoullisLawDerivationDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/BernoullisLawDerivationDiagram.svg/440px-BernoullisLawDerivationDiagram.svg.png 2x" data-file-width="790" data-file-height="370" /></a></span> <p><br /> </p> </td></tr> <tr> <td> <table class="mw-collapsible collapsed" style="width:100%"> <tbody><tr> <th scope="col" style="position:sticky; top:0; text-align: left; background-color: #87cefa;">法則 </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/%E8%B3%AA%E9%87%8F%E4%BF%9D%E5%AD%98%E3%81%AE%E6%B3%95%E5%89%87" title="質量保存の法則">質量保存の法則</a><br /><a href="/wiki/%E9%81%8B%E5%8B%95%E9%87%8F%E4%BF%9D%E5%AD%98%E3%81%AE%E6%B3%95%E5%89%87" class="mw-redirect" title="運動量保存の法則">運動量保存の法則</a><br /><a href="/wiki/%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC%E4%BF%9D%E5%AD%98%E3%81%AE%E6%B3%95%E5%89%87" title="エネルギー保存の法則">エネルギー保存の法則</a><br /><a href="/wiki/%E3%82%AF%E3%83%A9%E3%82%A6%E3%82%B8%E3%82%A6%E3%82%B9%E2%80%93%E3%83%87%E3%83%A5%E3%82%A8%E3%83%A0%E3%81%AE%E4%B8%8D%E7%AD%89%E5%BC%8F" title="クラウジウス–デュエムの不等式">クラウジウス–デュエムの不等式</a> </td></tr></tbody></table> <table class="mw-collapsible collapsed" style="width:100%"> <tbody><tr> <th scope="col" style="position:sticky; top:0; text-align: left; background-color: #87cefa;"><a href="/wiki/%E5%9B%BA%E4%BD%93%E5%8A%9B%E5%AD%A6" title="固体力学">固体力学</a> </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/%E5%9B%BA%E4%BD%93" title="固体">固体</a> · <a href="/wiki/%E5%A4%89%E5%BD%A2" title="変形">変形</a> · <a href="/wiki/%E5%BC%BE%E6%80%A7" title="弾性">弾性</a> · <a href="/wiki/%E5%BC%BE%E6%80%A7%E6%B3%A2" title="弾性波">弾性波</a> · <a href="/wiki/%E5%BC%BE%E5%A1%91%E6%80%A7" title="弾塑性">弾塑性</a> · <a href="/wiki/%E5%A1%91%E6%80%A7" title="塑性">塑性</a> · <a href="/wiki/%E3%83%95%E3%83%83%E3%82%AF%E3%81%AE%E6%B3%95%E5%89%87" title="フックの法則">フックの法則</a> · <a href="/wiki/%E5%BF%9C%E5%8A%9B" title="応力">応力</a> · <a href="/wiki/%E3%81%B2%E3%81%9A%E3%81%BF" title="ひずみ">ひずみ</a> · <a href="/wiki/%E6%9C%89%E9%99%90%E5%A4%89%E5%BD%A2%E7%90%86%E8%AB%96" title="有限変形理論">有限変形理論</a> · <a href="/wiki/%E3%83%AC%E3%82%AA%E3%83%AD%E3%82%B8%E3%83%BC" title="レオロジー">レオロジー</a> · <a href="/wiki/%E7%B2%98%E5%BC%BE%E6%80%A7" title="粘弾性">粘弾性</a> · <a href="/wiki/%E8%B6%85%E5%BC%BE%E6%80%A7" title="超弾性">超弾性</a> </td></tr></tbody></table> <table class="mw-collapsible collapsed" style="width:100%"> <tbody><tr> <th scope="col" style="position:sticky; top:0; text-align: left; background-color: #87cefa;"><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> </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/%E6%B5%81%E4%BD%93" title="流体">流体</a> · <a href="/wiki/%E6%B5%81%E4%BD%93%E9%9D%99%E5%8A%9B%E5%AD%A6" title="流体静力学">流体静力学</a><br /><a href="/wiki/%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6#概説" title="流体力学">流体動力学</a> · <a href="/wiki/%E7%B2%98%E5%BA%A6" title="粘度">粘度</a> · <a href="/wiki/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B5%81%E4%BD%93" title="ニュートン流体">ニュートン流体</a><br /><a href="/wiki/%E9%9D%9E%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E6%B5%81%E4%BD%93" title="非ニュートン流体">非ニュートン流体</a><br /><a href="/wiki/%E8%A1%A8%E9%9D%A2%E5%BC%B5%E5%8A%9B" title="表面張力">表面張力</a> </td></tr></tbody></table> <table class="mw-collapsible collapsed" style="width:100%"> <tbody><tr> <th scope="col" style="position:sticky; top:0; text-align: left; background-color: #87cefa;">科学者 </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/%E3%82%A2%E3%82%A4%E3%82%B6%E3%83%83%E3%82%AF%E3%83%BB%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3" title="アイザック・ニュートン">ニュートン</a> · <a href="/wiki/%E3%82%B8%E3%83%A7%E3%83%BC%E3%82%B8%E3%83%BB%E3%82%AC%E3%83%96%E3%83%AA%E3%82%A8%E3%83%AB%E3%83%BB%E3%82%B9%E3%83%88%E3%83%BC%E3%82%AF%E3%82%B9" title="ジョージ・ガブリエル・ストークス">ストークス</a> · <a href="/wiki/%E3%82%A2%E3%83%B3%E3%83%AA%E3%83%BB%E3%83%8A%E3%83%93%E3%82%A8" title="アンリ・ナビエ">ナビエ</a> · <a href="/wiki/%E3%82%AA%E3%83%BC%E3%82%AE%E3%83%A5%E3%82%B9%E3%82%BF%E3%83%B3%EF%BC%9D%E3%83%AB%E3%82%A4%E3%83%BB%E3%82%B3%E3%83%BC%E3%82%B7%E3%83%BC" title="オーギュスタン＝ルイ・コーシー">コーシー</a> · <a href="/wiki/%E3%83%AD%E3%83%90%E3%83%BC%E3%83%88%E3%83%BB%E3%83%95%E3%83%83%E3%82%AF" title="ロバート・フック">フック</a> · <a href="/wiki/%E3%83%80%E3%83%8B%E3%82%A8%E3%83%AB%E3%83%BB%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4" title="ダニエル・ベルヌーイ">ベルヌーイ</a> </td></tr></tbody></table> </td></tr> <tr style="text-align: center;"> <td><style data-mw-deduplicate="TemplateStyles:r99966302">.mw-parser-output .hlist ul,.mw-parser-output .hlist ol{padding-left:0}.mw-parser-output .hlist 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abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:88%}.mw-parser-output .navbox .navbar{display:block;font-size:88%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6" title="Template:連続体力学"><abbr title="参照先のページを表示します。">表</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template%E2%80%90%E3%83%8E%E3%83%BC%E3%83%88:%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6&action=edit&redlink=1" class="new" title="「Template‐ノート:連続体力学」 (存在しないページ)"><abbr title="参照先のノートを表示します。">話</abbr></a></li><li class="nv-edit"><a class="external text" href="https://ja.wikipedia.org/w/index.php?title=Template:%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6&action=edit"><abbr title="参照先のページを編集します。">編</abbr></a></li><li class="nv-hist"><a class="external text" href="https://ja.wikipedia.org/w/index.php?title=Template:%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6&action=history"><abbr title="参照先のページの履歴を表示します。">歴</abbr></a></li></ul></div> </td></tr></tbody></table> <p><b>物質微分</b>（ぶっしつびぶん、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">material derivative</span>）とは流れに乗って移動する<a href="/wiki/%E6%B5%81%E4%BD%93%E7%B2%92%E5%AD%90" title="流体粒子">流体粒子</a>の物理量 (<a href="/wiki/%E6%B8%A9%E5%BA%A6" title="温度">温度</a>や<a href="/wiki/%E9%81%8B%E5%8B%95%E9%87%8F" title="運動量">運動量</a>)の時間<a href="/wiki/%E5%A4%89%E5%8C%96%E7%8E%87" class="mw-redirect mw-disambig" title="変化率">変化率</a>のことで、<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6" title="連続体力学">連続体力学</a>の概念の一つである。固定された場所での物理量の時間変化でなく、流れに乗って動く仮想的な「観測者」が観た物理量の時間変化を記述する。 </p><p>物質微分は<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6#連続体の記述方法" title="連続体力学">ラグランジュ描像</a>に基づく時間変化を<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6#連続体の記述方法" title="連続体力学">オイラー描像</a>に基づく時間変化で記述したものである。物体固有の時間変化を記述するものなので物質微分 <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 \mathrm {D} /\mathrm {D} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} /\mathrm {D} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5fd0b46c79d746209affb6d5d41dd07e010f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.553ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} /\mathrm {D} t}"></span> は偏微分 <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 \partial /\partial t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial /\partial t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92814eb5d3b151bdb7851482e0704b34053dfdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle \partial /\partial t}"></span> と違い<span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=%E3%82%AC%E3%83%AA%E3%83%AC%E3%82%A4%E4%B8%8D%E5%A4%89&action=edit&redlink=1" class="new" title="「ガリレイ不変」 (存在しないページ)">ガリレイ不変</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/Galilean_invariance" class="extiw" title="en:Galilean invariance">英語版</a>）</span></span>である<sup id="cite_ref-吉澤流体_1-0" class="reference"><a href="#cite_note-吉澤流体-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p><p>名称としては他に、物質時間微分<sup id="cite_ref-連続体力学_2-0" class="reference"><a href="#cite_note-連続体力学-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>、流れに乗って移動するときの微分<sup id="cite_ref-日野流体_3-0" class="reference"><a href="#cite_note-日野流体-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>、実質微分<sup id="cite_ref-ハンドブック_4-0" class="reference"><a href="#cite_note-ハンドブック-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>、ラグランジュ微分<sup id="cite_ref-巽流体_5-0" class="reference"><a href="#cite_note-巽流体-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>などとも呼ばれる。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定義"><span id=".E5.AE.9A.E7.BE.A9"></span>定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=1" title="節を編集: 定義"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>速度場 <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 {\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2c2d3aac4213f3996d828c6aa8f4eb464a05cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {v}}}"></span> の流れにおける、<a href="/wiki/%E3%82%B9%E3%82%AB%E3%83%A9%E3%83%BC%E5%A0%B4" title="スカラー場">スカラー場</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 \varphi ({\boldsymbol {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ({\boldsymbol {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f056e75653c09dc5d40d46e7a48fd8559661be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.735ex; height:2.843ex;" alt="{\displaystyle \varphi ({\boldsymbol {x}},t)}"></span> および<a href="/wiki/%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E5%A0%B4" title="ベクトル場">ベクトル場</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 {\boldsymbol {A}}({\boldsymbol {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}({\boldsymbol {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f69087aa7f3378aba128bb631b998720c74d79f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.234ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {A}}({\boldsymbol {x}},t)}"></span> の物質微分は以下のように表される。 </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 {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}={\frac {\partial \varphi }{\partial t}}+{\boldsymbol {v}}\cdot \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}={\frac {\partial \varphi }{\partial t}}+{\boldsymbol {v}}\cdot \nabla \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb16bcb3d54b62f31dcce61124ecfd43d4b99c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.198ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}={\frac {\partial \varphi }{\partial t}}+{\boldsymbol {v}}\cdot \nabla \varphi }"></span></dd> <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 {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ba268b4d4b26d5070c01c4d72af018d0087ca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.696ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"></span></dd></dl> <p>ここで、それぞれの式の右辺第2項を<a href="/wiki/%E7%A7%BB%E6%B5%81" title="移流">移流</a>項、対流項と呼び、非一様な物理量の分布の中を移動したことで観測される物理量の変化率を表す。 </p> <div class="mw-heading mw-heading2"><h2 id="直観的意味"><span id=".E7.9B.B4.E8.A6.B3.E7.9A.84.E6.84.8F.E5.91.B3"></span>直観的意味</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=2" title="節を編集: 直観的意味"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>スカラー場<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 \varphi ({\boldsymbol {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ({\boldsymbol {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f056e75653c09dc5d40d46e7a48fd8559661be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.735ex; height:2.843ex;" alt="{\displaystyle \varphi ({\boldsymbol {x}},t)}"></span>の物質微分は直観的には流れに乗って動く物体から見た場合における<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 \varphi ({\boldsymbol {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ({\boldsymbol {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f056e75653c09dc5d40d46e7a48fd8559661be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.735ex; height:2.843ex;" alt="{\displaystyle \varphi ({\boldsymbol {x}},t)}"></span>の変化率を表す。（ベクトル場の場合も同様）。実際、位置のグラフ<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 {\boldsymbol {u}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a35d5ecef4ef5963d2d61d6dd557105d74cbec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.232ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {u}}(t)}"></span>で記述される<a href="/wiki/%E8%B3%AA%E7%82%B9" title="質点">質点</a>の軌道は速度場 <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 {\boldsymbol {v}}({\boldsymbol {x}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}({\boldsymbol {x}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa746aaab5e25ee20ea0af7a47601ab0dc0da3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.533ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {v}}({\boldsymbol {x}},t)}"></span> にそっているので、 </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 {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}={\boldsymbol {v}}({\boldsymbol {u}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}={\boldsymbol {v}}({\boldsymbol {u}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111ea3a178461859cd093adbb22cdf756107f3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.394ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}={\boldsymbol {v}}({\boldsymbol {u}},t)}"></span></dd></dl> <p>となるから(なお、この性質を満たす<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 {\boldsymbol {u}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a35d5ecef4ef5963d2d61d6dd557105d74cbec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.232ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {u}}(t)}"></span>を<a href="/wiki/%E6%B5%81%E8%B7%A1%E7%B7%9A" class="mw-redirect" title="流跡線">流跡線</a>という)、<a href="/wiki/%E3%83%A9%E3%82%A4%E3%83%97%E3%83%8B%E3%83%83%E3%83%84%E5%89%87" class="mw-redirect" title="ライプニッツ則">ライプニッツ則</a>から </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 {\frac {\mathrm {d} \varphi ({\boldsymbol {u}},t)}{\mathrm {d} t}}=(\nabla \varphi )\cdot {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}+{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\boldsymbol {v}}\cdot \nabla \varphi +{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \varphi ({\boldsymbol {u}},t)}{\mathrm {d} t}}=(\nabla \varphi )\cdot {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}+{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\boldsymbol {v}}\cdot \nabla \varphi +{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f635147c0ad536448168cadbf0eee165099c5b0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.48ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} \varphi ({\boldsymbol {u}},t)}{\mathrm {d} t}}=(\nabla \varphi )\cdot {\frac {\mathrm {d} {\boldsymbol {u}}}{\mathrm {d} t}}+{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\boldsymbol {v}}\cdot \nabla \varphi +{\frac {\mathrm {\partial } \varphi }{\mathrm {\partial } t}}={\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}"></span></dd></dl> <p>が成立する。上では一粒子しかない場合を想定したが、初期時刻における位置<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 {\boldsymbol {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/899e933d518eefcbbd0c48512cc7887ee117d040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.215ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {X}}}"></span>でパラメトライズされた粒子の族<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 {\boldsymbol {u}}({\boldsymbol {X}},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}({\boldsymbol {X}},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe9d4192ef2dbc5fc81ccf16857376f433be8f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.48ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {u}}({\boldsymbol {X}},t)}"></span>を考えた場合も、<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 {\boldsymbol {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/899e933d518eefcbbd0c48512cc7887ee117d040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.215ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {X}}}"></span>を固定して同様の証明を行う事で、同様の式が導ける。更に粒子の族を<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6" title="連続体力学">連続体</a>にまで拡張したものが物質微分である。 </p><p>以上の説明から分かるように、物質微分<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 {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe0b9f82a787d542426cc931dfef435c7255b6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.132ex; height:5.343ex;" alt="{\displaystyle {\frac {\mathrm {D} \varphi }{\mathrm {D} t}}}"></span>は物質に固定して観測する座標系（<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6#連続体の記述方法" title="連続体力学">物質表示</a>）における時間微分を表すが、それに対し通常の偏微分<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 {\frac {\partial \varphi }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1147c3bca8fc9edf3042971eeb6190ccbfb52bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.674ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}}"></span>は空間上に固定された座標系（<a href="/wiki/%E9%80%A3%E7%B6%9A%E4%BD%93%E5%8A%9B%E5%AD%A6#連続体の記述方法" title="連続体力学">空間表示</a>）における時間微分であるといえる。 </p> <div class="mw-heading mw-heading2"><h2 id="定常流"><span id=".E5.AE.9A.E5.B8.B8.E6.B5.81"></span>定常流</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=3" title="節を編集: 定常流"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%AE%9A%E5%B8%B8%E6%B5%81" title="定常流">定常流</a>はすべての物理量のオイラー描像的時間変化率が <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 {\partial /\partial t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial /\partial t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0437cb73c5e613c0086543a152bab9cf8687acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.899ex; height:2.843ex;" alt="{\displaystyle {\partial /\partial t}=0}"></span> となる流れであるが、ラグランジュ描像的時間変化率が <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 {\mathrm {D} /\mathrm {D} t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {D} /\mathrm {D} t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bae67b0e0f6fd0027e058294e240b1cca74850c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.814ex; height:2.843ex;" alt="{\displaystyle {\mathrm {D} /\mathrm {D} t}=0}"></span> となるとは限らないことに注意すべきである。 </p><p>一つの<a href="/wiki/%E6%B5%81%E7%B7%9A" title="流線">流線</a>に着目する。流線上のある点からの道のりを <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> 、流線の単位接ベクトルを <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 {\boldsymbol {e}}_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5692239f48052d3706bd5e6b3996623483078b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {e}}_{s}}"></span> と表す。速度ベクトルは流線に接しているので、定常流における物質微分は </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 {\begin{aligned}{\mathrm {D} \over \mathrm {D} t}&={\partial \over \partial t}+{\boldsymbol {v}}\cdot \nabla \\&=0+(v{\boldsymbol {e}}_{s})\cdot \nabla \\&=v{\partial \over \partial s}&(\because ~{\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>v</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo stretchy="false">(</mo> <mo>∵</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathrm {D} \over \mathrm {D} t}&={\partial \over \partial t}+{\boldsymbol {v}}\cdot \nabla \\&=0+(v{\boldsymbol {e}}_{s})\cdot \nabla \\&=v{\partial \over \partial s}&(\because ~{\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b363c222770800e3a773536d7dc43b385a6206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.332ex; margin-bottom: -0.172ex; width:41.627ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\mathrm {D} \over \mathrm {D} t}&={\partial \over \partial t}+{\boldsymbol {v}}\cdot \nabla \\&=0+(v{\boldsymbol {e}}_{s})\cdot \nabla \\&=v{\partial \over \partial s}&(\because ~{\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s})\end{aligned}}}"></span></dd></dl> <p>となり、流線方向の変化率に速さをかけたものに等しいことが導かれる。 </p><p>これから、定常流( <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 {\partial /\partial t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial /\partial t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0437cb73c5e613c0086543a152bab9cf8687acb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.899ex; height:2.843ex;" alt="{\displaystyle {\partial /\partial t}=0}"></span> )でも、流線に沿って物理量が変化するなら<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 {\mathrm {D} /\mathrm {D} t}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {D} /\mathrm {D} t}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e50e178f2fe6db83b57edd1d5eb31a6a7749f27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.814ex; height:2.843ex;" alt="{\displaystyle {\mathrm {D} /\mathrm {D} t}\neq 0}"></span> であることがわかる。 </p> <div class="mw-heading mw-heading3"><h3 id="定常流における加速度"><span id=".E5.AE.9A.E5.B8.B8.E6.B5.81.E3.81.AB.E3.81.8A.E3.81.91.E3.82.8B.E5.8A.A0.E9.80.9F.E5.BA.A6"></span>定常流における加速度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=4" title="節を編集: 定常流における加速度"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>応用で重要なのは速度の物質微分すなわち加速度である。定常流、つまり、速度の時間変化がない流れでも、流体粒子の加速度は0とは限らない。定常流でも、流線に沿って速度の大きさは変化しうるし、流線に沿って速度の方向が変わる（流線が曲がる）こともありうる。これを式に表すと、 </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 {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117edcf9ce3f78e8da214eeee9f986bd541129ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.898ex; height:6.343ex;" alt="{\displaystyle {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}}"></span></dd></dl> <p>ただし、 <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> は流線上のある点からの道のり、<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> は瞬間的な<a href="/wiki/%E6%9B%B2%E7%8E%87%E4%B8%AD%E5%BF%83" class="mw-redirect" title="曲率中心">曲率中心</a>からの距離、<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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> は流線の<a href="/wiki/%E6%9B%B2%E7%8E%87%E5%8D%8A%E5%BE%84" class="mw-redirect" title="曲率半径">曲率半径</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 {\boldsymbol {e}}_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5692239f48052d3706bd5e6b3996623483078b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {e}}_{s}}"></span> は接線方向の単位ベクトル、<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 {\boldsymbol {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2468e50b3af4f15b2a70efc121995106d25788a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {e}}_{r}}"></span> は半径方向の単位ベクトルを表す。 </p> <dl><dd><table class="toccolours mw-collapsible mw-collapsed" width="60%" style="text-align:left"> <tbody><tr> <th>導出 </th></tr> <tr> <td> <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 {\begin{aligned}{\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}&={\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&=v{\partial (v{\boldsymbol {e}}_{s}) \over \partial s}\\&=v{\partial v \over \partial s}{\boldsymbol {e}}_{s}+v^{2}{\partial {\boldsymbol {e}}_{s} \over \partial s}\\&={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>v</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}&={\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&=v{\partial (v{\boldsymbol {e}}_{s}) \over \partial s}\\&=v{\partial v \over \partial s}{\boldsymbol {e}}_{s}+v^{2}{\partial {\boldsymbol {e}}_{s} \over \partial s}\\&={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7383be9c4069457a7077a670592af9da7e06c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:28.649ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}{\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}&={\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&=v{\partial (v{\boldsymbol {e}}_{s}) \over \partial s}\\&=v{\partial v \over \partial s}{\boldsymbol {e}}_{s}+v^{2}{\partial {\boldsymbol {e}}_{s} \over \partial s}\\&={\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}\end{aligned}}}"></span></dd></dl> <p>ただし、曲線の<a href="/wiki/%E6%9B%B2%E7%8E%87" title="曲率">曲率</a>についての関係式 </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 {\partial {\boldsymbol {e}}_{s} \over \partial s}=-{1 \over R}{\boldsymbol {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial {\boldsymbol {e}}_{s} \over \partial s}=-{1 \over R}{\boldsymbol {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635e3f0c99e7cf2d5cca7a90148f47b2cb0b0337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.214ex; height:5.509ex;" alt="{\displaystyle {\partial {\boldsymbol {e}}_{s} \over \partial s}=-{1 \over R}{\boldsymbol {e}}_{r}}"></span></dd></dl> <p>を使った。 </p> </td></tr></tbody></table></dd></dl> <p>加速度の流線方向の成分は流線にそった速さの変化率に対応し、加速度の法線方向の成分は流線が曲がることによる向心加速度に対応する。 </p> <div class="mw-heading mw-heading3"><h3 id="ベルヌーイの定理と流線曲率の定理"><span id=".E3.83.99.E3.83.AB.E3.83.8C.E3.83.BC.E3.82.A4.E3.81.AE.E5.AE.9A.E7.90.86.E3.81.A8.E6.B5.81.E7.B7.9A.E6.9B.B2.E7.8E.87.E3.81.AE.E5.AE.9A.E7.90.86"></span>ベルヌーイの定理と流線曲率の定理</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=5" title="節を編集: ベルヌーイの定理と流線曲率の定理"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>外力のない<a href="/wiki/%E9%9D%9E%E7%B2%98%E6%80%A7" class="mw-redirect" title="非粘性">非粘性</a>の<a href="/wiki/%E3%83%90%E3%83%AD%E3%83%88%E3%83%AD%E3%83%94%E3%83%83%E3%82%AF%E6%B5%81%E4%BD%93" class="mw-redirect" title="バロトロピック流体">バロトロピック流体</a>の定常な流れを考える。非粘性流体の流れを記述する<a href="/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E6%96%B9%E7%A8%8B%E5%BC%8F_(%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6)" title="オイラー方程式 (流体力学)">オイラー方程式</a> </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 {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}=-{1 \over \rho }\nabla p+{\boldsymbol {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}=-{1 \over \rho }\nabla p+{\boldsymbol {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32379b573981a1c54761a9178b474297e26600f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.271ex; height:5.676ex;" alt="{\displaystyle {\mathrm {D} {\boldsymbol {v}} \over \mathrm {D} t}=-{1 \over \rho }\nabla p+{\boldsymbol {f}}}"></span></dd></dl> <p>は定常、外力がない、バロトロピックという条件では </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 {\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}=-\nabla \int {\mathrm {d} p \over \rho }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}=-\nabla \int {\mathrm {d} p \over \rho }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f860f98c27b178a894faeab6907470ceef7742b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.978ex; height:6.343ex;" alt="{\displaystyle {\partial \over \partial s}\left({v^{2} \over 2}\right){\boldsymbol {e}}_{s}-{v^{2} \over R}{\boldsymbol {e}}_{r}=-\nabla \int {\mathrm {d} p \over \rho }}"></span></dd></dl> <p>と変形できる。 </p><p>方程式の両辺にそれぞれ <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 {\boldsymbol {e}}_{s},\,{\boldsymbol {e}}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}_{s},\,{\boldsymbol {e}}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7e779a7a792cd14430e6339c07f464e3ed7003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.974ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {e}}_{s},\,{\boldsymbol {e}}_{r}}"></span> を内積でかけることで、流線方向(接線)成分、半径方向（主法線）成分は、 </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 {\begin{aligned}{\partial \over \partial s}\left({v^{2} \over 2}\right)&=-{\partial \over \partial s}\int {\mathrm {d} p \over \rho }\qquad \quad \therefore ~{\partial \over \partial s}\left({v^{2} \over 2}+\int {\mathrm {d} p \over \rho }\right)=0\\-{v^{2} \over R}&=-\left.{\partial \over \partial r}\int {\mathrm {d} p \over \rho }\right|_{r=R}\quad \therefore ~{\partial p \over \partial r}=\rho {v^{2} \over r}\quad \left(\mathrm {or~} {\partial \over \partial r}\int {\mathrm {d} p \over \rho }={v^{2} \over r}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mspace width="2em" /> <mspace width="1em" /> <mo>∴</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mi>R</mi> </mrow> </msub> <mspace width="1em" /> <mo>∴</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\partial \over \partial s}\left({v^{2} \over 2}\right)&=-{\partial \over \partial s}\int {\mathrm {d} p \over \rho }\qquad \quad \therefore ~{\partial \over \partial s}\left({v^{2} \over 2}+\int {\mathrm {d} p \over \rho }\right)=0\\-{v^{2} \over R}&=-\left.{\partial \over \partial r}\int {\mathrm {d} p \over \rho }\right|_{r=R}\quad \therefore ~{\partial p \over \partial r}=\rho {v^{2} \over r}\quad \left(\mathrm {or~} {\partial \over \partial r}\int {\mathrm {d} p \over \rho }={v^{2} \over r}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0215ba4a65250b53b388c47f69e0154e191c45a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:70.101ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\partial \over \partial s}\left({v^{2} \over 2}\right)&=-{\partial \over \partial s}\int {\mathrm {d} p \over \rho }\qquad \quad \therefore ~{\partial \over \partial s}\left({v^{2} \over 2}+\int {\mathrm {d} p \over \rho }\right)=0\\-{v^{2} \over R}&=-\left.{\partial \over \partial r}\int {\mathrm {d} p \over \rho }\right|_{r=R}\quad \therefore ~{\partial p \over \partial r}=\rho {v^{2} \over r}\quad \left(\mathrm {or~} {\partial \over \partial r}\int {\mathrm {d} p \over \rho }={v^{2} \over r}\right)\end{aligned}}}"></span></dd></dl> <p>と表せる。ただし、<a href="/wiki/%E6%96%B9%E5%90%91%E5%BE%AE%E5%88%86" title="方向微分">方向微分</a>の性質： </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 {\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s},\,{\boldsymbol {e}}_{r}\cdot \nabla ={\partial \over \partial r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s},\,{\boldsymbol {e}}_{r}\cdot \nabla ={\partial \over \partial r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/712fe326c458c1010740a7ea204282b1888c5100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.848ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {e}}_{s}\cdot \nabla ={\partial \over \partial s},\,{\boldsymbol {e}}_{r}\cdot \nabla ={\partial \over \partial r}}"></span></dd></dl> <p>を使った。 </p><p>第1式が<a href="/wiki/%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E3%81%AE%E5%AE%9A%E7%90%86" title="ベルヌーイの定理">ベルヌーイの定理</a>、第2式が<a href="/wiki/%E6%B5%81%E7%B7%9A%E6%9B%B2%E7%8E%87%E3%81%AE%E5%AE%9A%E7%90%86" title="流線曲率の定理">流線曲率の定理</a>に対応する。 </p> <div class="mw-heading mw-heading2"><h2 id="対流項"><span id=".E5.AF.BE.E6.B5.81.E9.A0.85"></span>対流項</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=6" title="節を編集: 対流項"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>移流項における <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 \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.456ex; height:2.676ex;" alt="{\displaystyle \nabla \varphi }"></span> はスカラー量の<a href="/wiki/%E5%8B%BE%E9%85%8D" title="勾配">勾配</a>であるが、対流項における <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 \nabla {\boldsymbol {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla {\boldsymbol {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ae2eabcc09ac9f8c43109052822ce02f429ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.955ex; height:2.176ex;" alt="{\displaystyle \nabla {\boldsymbol {A}}}"></span> はベクトル量の<a href="/wiki/%E3%82%AF%E3%83%AA%E3%82%B9%E3%83%88%E3%83%83%E3%83%95%E3%82%A7%E3%83%AB%E8%A8%98%E5%8F%B7#概要" title="クリストッフェル記号">共変微分</a>である。ベクトル量の対流項 <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 {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750465fe3e55c1a8ed7a164b18c853b6506b1b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.953ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"></span> を <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 ({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi>grad</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ab20d13ccf52cf7f2a36864de8016739a97a31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.355ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}}"></span> と記述することがあるが、この表示は<a href="/wiki/%E3%83%87%E3%82%AB%E3%83%AB%E3%83%88%E5%BA%A7%E6%A8%99%E7%B3%BB" class="mw-redirect" title="デカルト座標系">デカルト座標系</a>でしか等価でないことに注意すべきである<sup id="cite_ref-巽流体_5-1" class="reference"><a href="#cite_note-巽流体-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>（スカラー量の対流項 <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 {\boldsymbol {v}}\cdot \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}\cdot \nabla \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934c8a1a18ffdc4cddace407f8e30e243cc38707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.453ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {v}}\cdot \nabla \varphi }"></span> については <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 ({\boldsymbol {v}}\cdot \operatorname {grad} )\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi>grad</mi> <mo stretchy="false">)</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {v}}\cdot \operatorname {grad} )\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4498dcaf7c1692ef625dc4ac18c3f4b972182b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.856ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {v}}\cdot \operatorname {grad} )\varphi }"></span> と等価である）。 </p><p>共変微分を使わずに一般の座標系で成り立つ表現としては<sup id="cite_ref-巽流体_5-2" class="reference"><a href="#cite_note-巽流体-5"><span class="cite-bracket">[</span>5<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 {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\frac {1}{2}}\left\{\operatorname {grad} ({\boldsymbol {v}}\cdot {\boldsymbol {A}})+\operatorname {rot} {\boldsymbol {v}}\times {\boldsymbol {A}}+\operatorname {rot} {\boldsymbol {A}}\times {\boldsymbol {v}}-\operatorname {rot} ({\boldsymbol {v}}\times {\boldsymbol {A}})+{\boldsymbol {v}}\,\operatorname {div} {\boldsymbol {A}}-{\boldsymbol {A}}\,\operatorname {div} {\boldsymbol {v}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mi>grad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo>+</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>−</mo> <mi>rot</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mspace width="thinmathspace" /> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mspace width="thinmathspace" /> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\frac {1}{2}}\left\{\operatorname {grad} ({\boldsymbol {v}}\cdot {\boldsymbol {A}})+\operatorname {rot} {\boldsymbol {v}}\times {\boldsymbol {A}}+\operatorname {rot} {\boldsymbol {A}}\times {\boldsymbol {v}}-\operatorname {rot} ({\boldsymbol {v}}\times {\boldsymbol {A}})+{\boldsymbol {v}}\,\operatorname {div} {\boldsymbol {A}}-{\boldsymbol {A}}\,\operatorname {div} {\boldsymbol {v}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3b59ef6da7b7b65a807b23657ef6b56600a780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:90.396ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {D} {\boldsymbol {A}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {A}}}{\partial t}}+{\frac {1}{2}}\left\{\operatorname {grad} ({\boldsymbol {v}}\cdot {\boldsymbol {A}})+\operatorname {rot} {\boldsymbol {v}}\times {\boldsymbol {A}}+\operatorname {rot} {\boldsymbol {A}}\times {\boldsymbol {v}}-\operatorname {rot} ({\boldsymbol {v}}\times {\boldsymbol {A}})+{\boldsymbol {v}}\,\operatorname {div} {\boldsymbol {A}}-{\boldsymbol {A}}\,\operatorname {div} {\boldsymbol {v}}\right\}}"></span></dd></dl> <p>がある。特に<a href="/wiki/%E5%8A%A0%E9%80%9F%E5%BA%A6" title="加速度">加速度</a>の回転形表示 </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 {\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>grad</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>×</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628bb6676292811002b77864bacb4bf3ed9980d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.736ex; height:7.509ex;" alt="{\displaystyle {\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}}"></span></dd></dl> <p>は重要である。 </p> <dl><dd><table class="toccolours mw-collapsible mw-collapsed" width="60%" style="text-align:left"> <tbody><tr> <th>エディントンのイプシロンを用いた導出 </th></tr> <tr> <td> <p><a href="/wiki/%E3%82%A8%E3%83%87%E3%82%A3%E3%83%B3%E3%83%88%E3%83%B3%E3%81%AE%E3%82%A4%E3%83%97%E3%82%B7%E3%83%AD%E3%83%B3" title="エディントンのイプシロン">エディントンのイプシロン</a>の性質 </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 {\begin{aligned}({\boldsymbol {a}}\times {\boldsymbol {b}})_{i}&=\sum _{jk}\varepsilon _{ijk}a_{j}b_{k}\\\sum _{k}\varepsilon _{ijk}\varepsilon _{k\ell m}&=\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">b</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}({\boldsymbol {a}}\times {\boldsymbol {b}})_{i}&=\sum _{jk}\varepsilon _{ijk}a_{j}b_{k}\\\sum _{k}\varepsilon _{ijk}\varepsilon _{k\ell m}&=\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b8790842528ac31b14672c052f9dd94b46d072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.963ex; margin-bottom: -0.208ex; width:29.955ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}({\boldsymbol {a}}\times {\boldsymbol {b}})_{i}&=\sum _{jk}\varepsilon _{ijk}a_{j}b_{k}\\\sum _{k}\varepsilon _{ijk}\varepsilon _{k\ell m}&=\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell }\end{aligned}}}"></span></dd></dl> <p>を使えば、 </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 {\begin{aligned}({\boldsymbol {v}}\times \operatorname {rot} \,{\boldsymbol {v}})_{i}&=\sum _{jk}\varepsilon _{ijk}v_{j}(\operatorname {rot} \,{\boldsymbol {v}})_{k}\\&=\sum _{jk}\varepsilon _{ijk}v_{j}\sum _{\ell m}\varepsilon _{k\ell m}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{j\ell m}(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })v_{j}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{m}\left(v_{m}{\partial \over \partial x_{i}}v_{m}-v_{m}{\partial \over \partial x_{m}}v_{i}\right)\\&=\left(\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\right)_{i}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>×</mo> <mi>rot</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>rot</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mi>m</mi> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> <mi>m</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow> <mi>grad</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}({\boldsymbol {v}}\times \operatorname {rot} \,{\boldsymbol {v}})_{i}&=\sum _{jk}\varepsilon _{ijk}v_{j}(\operatorname {rot} \,{\boldsymbol {v}})_{k}\\&=\sum _{jk}\varepsilon _{ijk}v_{j}\sum _{\ell m}\varepsilon _{k\ell m}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{j\ell m}(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })v_{j}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{m}\left(v_{m}{\partial \over \partial x_{i}}v_{m}-v_{m}{\partial \over \partial x_{m}}v_{i}\right)\\&=\left(\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\right)_{i}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba2171219e15dd478f78d5e9b9002701ec02b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.447ex; margin-bottom: -0.224ex; width:45.47ex; height:34.509ex;" alt="{\displaystyle {\begin{aligned}({\boldsymbol {v}}\times \operatorname {rot} \,{\boldsymbol {v}})_{i}&=\sum _{jk}\varepsilon _{ijk}v_{j}(\operatorname {rot} \,{\boldsymbol {v}})_{k}\\&=\sum _{jk}\varepsilon _{ijk}v_{j}\sum _{\ell m}\varepsilon _{k\ell m}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{j\ell m}(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })v_{j}{\partial \over \partial x_{\ell }}v_{m}\\&=\sum _{m}\left(v_{m}{\partial \over \partial x_{i}}v_{m}-v_{m}{\partial \over \partial x_{m}}v_{i}\right)\\&=\left(\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\right)_{i}\end{aligned}}}"></span></dd></dl> <p>より、 </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 {\begin{aligned}{\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>grad</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>×</mo> <mi>rot</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923946da2104514600a1fe04b49c57da2c8f8b17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.752ex; margin-bottom: -0.253ex; width:38.488ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {D} {\boldsymbol {v}}}{\mathrm {D} t}}&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+{\boldsymbol {v}}\cdot \nabla {\boldsymbol {v}}\\&={\frac {\partial {\boldsymbol {v}}}{\partial t}}+\operatorname {grad} \left({\frac {|{\boldsymbol {v}}|^{2}}{2}}\right)-{\boldsymbol {v}}\times \operatorname {rot} {\boldsymbol {v}}\end{aligned}}}"></span></dd></dl> <p>が得られる。 </p> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="曲線直交座標系"><span id=".E6.9B.B2.E7.B7.9A.E7.9B.B4.E4.BA.A4.E5.BA.A7.E6.A8.99.E7.B3.BB"></span>曲線直交座標系</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=7" title="節を編集: 曲線直交座標系"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>曲線直交座標系 <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 {\boldsymbol {r}}={\boldsymbol {r}}(q^{1},q^{2},q^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {r}}={\boldsymbol {r}}(q^{1},q^{2},q^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c281d16bc19f36bdcf886e23bf8b6a9c981fae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.836ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {r}}={\boldsymbol {r}}(q^{1},q^{2},q^{3})}"></span> における対流項 <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 {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750465fe3e55c1a8ed7a164b18c853b6506b1b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.953ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}}"></span> の <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> 成分は以下のように与えられる<sup id="cite_ref-mathworld_6-0" class="reference"><a href="#cite_note-mathworld-6"><span class="cite-bracket">[</span>6<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 [{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}]_{j}=\sum _{k}\left\{{\frac {v_{k}}{h_{k}}}{\frac {\partial A_{j}}{\partial q^{k}}}+{\frac {A_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}]_{j}=\sum _{k}\left\{{\frac {v_{k}}{h_{k}}}{\frac {\partial A_{j}}{\partial q^{k}}}+{\frac {A_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/548e429331d003aea1510e1d6fe56ac890b86c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.794ex; height:6.843ex;" alt="{\displaystyle [{\boldsymbol {v}}\cdot \nabla {\boldsymbol {A}}]_{j}=\sum _{k}\left\{{\frac {v_{k}}{h_{k}}}{\frac {\partial A_{j}}{\partial q^{k}}}+{\frac {A_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}"></span></dd></dl> <p>ただし、 </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 h_{k}=\left|{\partial {\boldsymbol {r}} \over \partial q^{k}}\right|={\sqrt {g_{kk}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{k}=\left|{\partial {\boldsymbol {r}} \over \partial q^{k}}\right|={\sqrt {g_{kk}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d312fc7c74f527e25aef01722d0d7f7322d817f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.231ex; height:6.176ex;" alt="{\displaystyle h_{k}=\left|{\partial {\boldsymbol {r}} \over \partial q^{k}}\right|={\sqrt {g_{kk}}}}"></span></dd></dl> <p>（<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 g_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c1130c3dec178129b287a3672c72f88e773832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}"></span> は<a href="/wiki/%E8%A8%88%E9%87%8F%E3%83%86%E3%83%B3%E3%82%BD%E3%83%AB" title="計量テンソル">計量テンソル</a>）である。 </p><p>先で述べたように </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 [({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}]_{j}=\sum _{k}v_{k}\left({\frac {1}{h_{k}}}{\frac {\partial }{\partial q^{k}}}\right)A_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>⋅</mo> <mi>grad</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}]_{j}=\sum _{k}v_{k}\left({\frac {1}{h_{k}}}{\frac {\partial }{\partial q^{k}}}\right)A_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32381b8596d11aca67c542d1cbe5cbb6d5f01de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.05ex; height:6.676ex;" alt="{\displaystyle [({\boldsymbol {v}}\cdot \operatorname {grad} ){\boldsymbol {A}}]_{j}=\sum _{k}v_{k}\left({\frac {1}{h_{k}}}{\frac {\partial }{\partial q^{k}}}\right)A_{j}}"></span></dd></dl> <p>とはデカルト座標系 <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 (h_{1}=h_{2}=h_{3}=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h_{1}=h_{2}=h_{3}=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef7d96ce1f765f50c7d6951914468c3465b3c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.447ex; height:2.843ex;" alt="{\displaystyle (h_{1}=h_{2}=h_{3}=1)}"></span> においてのみ等しい。 </p><p><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 {\boldsymbol {A}}={\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}={\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c28e51cd7886c1849a95886525aaa92a9b7db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.436ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}={\boldsymbol {v}}}"></span> とした時の物質微分(＝加速度)の対流項に現れる第2項 </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 \sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e37ac9cf6812bebc6b35a0e058fa7961eccc5c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.628ex; height:6.843ex;" alt="{\displaystyle \sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\frac {\partial h_{j}}{\partial q^{k}}}-v_{k}{\frac {\partial h_{k}}{\partial q^{j}}}\right)\right\}}"></span></dd></dl> <p>は曲線直交座標系で現れる<a href="/w/index.php?title=%E8%A6%8B%E3%81%8B%E3%81%91%E3%81%AE%E5%8A%9B&action=edit&redlink=1" class="new" title="「見かけの力」 (存在しないページ)">見かけの力</a>に対応する。 </p><p>実際、 <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 L={\frac {m}{2}}\sum _{k}(h_{k}{\dot {q}}^{k})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {m}{2}}\sum _{k}(h_{k}{\dot {q}}^{k})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358ad5e1ef0df366dfad5a50879ea5b683337900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.057ex; height:5.843ex;" alt="{\displaystyle L={\frac {m}{2}}\sum _{k}(h_{k}{\dot {q}}^{k})^{2}}"></span> に対して <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 ({\mathrm {d} /\mathrm {d} t})({\partial L/\partial {\dot {q}}^{j}})-({\partial L/\partial q^{j}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathrm {d} /\mathrm {d} t})({\partial L/\partial {\dot {q}}^{j}})-({\partial L/\partial q^{j}})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76b419f6707b916ad1d466fa717d6841466dae38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.155ex; height:3.176ex;" alt="{\displaystyle ({\mathrm {d} /\mathrm {d} t})({\partial L/\partial {\dot {q}}^{j}})-({\partial L/\partial q^{j}})=0}"></span> を計算すると、 </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 {\mathrm {d} v_{j} \over \mathrm {d} t}-\sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\partial h_{j} \over \partial q^{k}}-v_{k}{\partial h_{k} \over \partial q^{j}}\right)\right\}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathrm {d} v_{j} \over \mathrm {d} t}-\sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\partial h_{j} \over \partial q^{k}}-v_{k}{\partial h_{k} \over \partial q^{j}}\right)\right\}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702022ae1b96e3678a6f8fa8aad681fb05152bd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.895ex; height:6.843ex;" alt="{\displaystyle {\mathrm {d} v_{j} \over \mathrm {d} t}-\sum _{k}\left\{{\frac {v_{k}}{h_{k}h_{j}}}\left(v_{j}{\partial h_{j} \over \partial q^{k}}-v_{k}{\partial h_{k} \over \partial q^{j}}\right)\right\}=0}"></span></dd></dl> <p>が得られる。ただし、<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 v_{k}=h_{k}{\dot {q}}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{k}=h_{k}{\dot {q}}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd40c4fdb5ff36a588c2ddcf13f1e3b83c2dad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.208ex; height:3.009ex;" alt="{\displaystyle v_{k}=h_{k}{\dot {q}}^{k}}"></span> であり、 <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 {\dot {h}}_{k}=\sum _{i}{\partial h_{k} \over \partial q^{i}}{\dot {q}}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>h</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {h}}_{k}=\sum _{i}{\partial h_{k} \over \partial q^{i}}{\dot {q}}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f902ed93e8c2d4a57252d07084b4f604fcda326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.027ex; height:6.509ex;" alt="{\displaystyle {\dot {h}}_{k}=\sum _{i}{\partial h_{k} \over \partial q^{i}}{\dot {q}}^{i}}"></span> を使う。 </p> <div class="mw-heading mw-heading2"><h2 id="相対論的物質微分"><span id=".E7.9B.B8.E5.AF.BE.E8.AB.96.E7.9A.84.E7.89.A9.E8.B3.AA.E5.BE.AE.E5.88.86"></span>相対論的物質微分</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=8" title="節を編集: 相対論的物質微分"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>時間<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>の代わりに固有時間<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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>による物質微分を構成することもできる．具体的に，次式で与えられる<sup class="noprint Template-Fact">[<i><a href="/wiki/Wikipedia:%E3%80%8C%E8%A6%81%E5%87%BA%E5%85%B8%E3%80%8D%E3%82%92%E3%82%AF%E3%83%AA%E3%83%83%E3%82%AF%E3%81%95%E3%82%8C%E3%81%9F%E6%96%B9%E3%81%B8" title="Wikipedia:「要出典」をクリックされた方へ"><span title="この記述には信頼できる情報源の提示が求められています。（2016年5月）">要出典</span></a></i>]</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 {\frac {\mathrm {D} A^{\mu }(x)}{\mathrm {D} \tau }}=v^{\nu }\nabla _{\nu }A^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} A^{\mu }(x)}{\mathrm {D} \tau }}=v^{\nu }\nabla _{\nu }A^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6c2a36cb81df6c1e475c898cc6277ebaba4fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.7ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathrm {D} A^{\mu }(x)}{\mathrm {D} \tau }}=v^{\nu }\nabla _{\nu }A^{\mu }.}"></span></dd></dl> <p><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 v^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932f59612f0421be71d9f6ee58a7ef4417f2d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.231ex; height:2.343ex;" alt="{\displaystyle v^{\nu }}"></span>は流体の四元速度，<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 A^{\mu }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f880877b6b53b2e1d7be64ad5ffe3267fb8720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.105ex; height:2.843ex;" alt="{\displaystyle A^{\mu }(x)}"></span>は四元時空を変数とするベクトルである．とくに，<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 A^{\mu }(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f880877b6b53b2e1d7be64ad5ffe3267fb8720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.105ex; height:2.843ex;" alt="{\displaystyle A^{\mu }(x)}"></span>を速度とすると，測地線方程式となり，零になる． </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 {\begin{aligned}{\frac {\mathrm {D} v^{\mu }(x)}{\mathrm {D} \tau }}&=v^{\nu }\nabla _{\nu }v^{\mu }=v^{\nu }\left(\partial _{\nu }v^{\mu }+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }\right)\\&={\frac {\partial x^{\nu }}{\partial \tau }}{\frac {\partial v^{\mu }}{\partial x^{\nu }}}+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }v^{\nu }\\&={\frac {\partial ^{2}x^{\mu }}{\partial \tau ^{2}}}+\Gamma _{\nu \lambda }^{\mu }{\frac {\partial x^{\lambda }}{\partial \tau }}{\frac {\partial x^{\lambda }}{\partial \tau }}=0.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>τ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {D} v^{\mu }(x)}{\mathrm {D} \tau }}&=v^{\nu }\nabla _{\nu }v^{\mu }=v^{\nu }\left(\partial _{\nu }v^{\mu }+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }\right)\\&={\frac {\partial x^{\nu }}{\partial \tau }}{\frac {\partial v^{\mu }}{\partial x^{\nu }}}+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }v^{\nu }\\&={\frac {\partial ^{2}x^{\mu }}{\partial \tau ^{2}}}+\Gamma _{\nu \lambda }^{\mu }{\frac {\partial x^{\lambda }}{\partial \tau }}{\frac {\partial x^{\lambda }}{\partial \tau }}=0.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82414ee6bdaa2fa62cbd92f8320c2c7a67e2741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.924ex; margin-bottom: -0.247ex; width:40.782ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {D} v^{\mu }(x)}{\mathrm {D} \tau }}&=v^{\nu }\nabla _{\nu }v^{\mu }=v^{\nu }\left(\partial _{\nu }v^{\mu }+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }\right)\\&={\frac {\partial x^{\nu }}{\partial \tau }}{\frac {\partial v^{\mu }}{\partial x^{\nu }}}+\Gamma _{\nu \lambda }^{\mu }v^{\lambda }v^{\nu }\\&={\frac {\partial ^{2}x^{\mu }}{\partial \tau ^{2}}}+\Gamma _{\nu \lambda }^{\mu }{\frac {\partial x^{\lambda }}{\partial \tau }}{\frac {\partial x^{\lambda }}{\partial \tau }}=0.\\\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="脚注"><span id=".E8.84.9A.E6.B3.A8"></span>脚注</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=9" title="節を編集: 脚注"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" style="float:right; font-size:90%;">[<a href="/wiki/Help:%E8%84%9A%E6%B3%A8/%E8%AA%AD%E8%80%85%E5%90%91%E3%81%91" title="Help:脚注/読者向け"><span title="この欄の操作法">脚注の使い方</span></a>]</div> <div class="mw-heading mw-heading3"><h3 id="出典"><span id=".E5.87.BA.E5.85.B8"></span>出典</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=10" title="節を編集: 出典"><span>編集</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-吉澤流体-1"><b><a href="#cite_ref-吉澤流体_1-0">^</a></b> <span class="reference-text">吉澤徴『流体力学』東京大学出版、2001年9月6日初版発行、<a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4130626035" class="internal mw-magiclink-isbn">ISBN 4130626035</a></span> </li> <li id="cite_note-連続体力学-2"><b><a href="#cite_ref-連続体力学_2-0">^</a></b> <span class="reference-text">田村武『連続体力学入門』朝倉書店、2000年2月20日初版１刷発行、<a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4254201028" class="internal mw-magiclink-isbn">ISBN 4254201028</a></span> </li> <li id="cite_note-日野流体-3"><b><a href="#cite_ref-日野流体_3-0">^</a></b> <span class="reference-text">日野幹雄『流体力学』朝倉書店、1992年12月10日初版１刷発行、<a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4254200668" class="internal mw-magiclink-isbn">ISBN 4254200668</a></span> </li> <li id="cite_note-ハンドブック-4"><b><a href="#cite_ref-ハンドブック_4-0">^</a></b> <span class="reference-text">中村育雄『流体解析ハンドブック』共立出版、1998年3月20日初版１刷発行、<a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/4320081188" class="internal mw-magiclink-isbn">ISBN 4320081188</a></span> </li> <li id="cite_note-巽流体-5">^ <a href="#cite_ref-巽流体_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-巽流体_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-巽流体_5-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text">巽友正　『新物理学シリーズ21 流体力学』　培風館、1982年 4月15日初版発行、<a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/456302421X" class="internal mw-magiclink-isbn">ISBN 4-563-02421-X</a></span> </li> <li id="cite_note-mathworld-6"><b><a href="#cite_ref-mathworld_6-0">^</a></b> <span class="reference-text">Eric W. Weisstein "Convective Operator" <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/ConvectiveOperator.html">http://mathworld.wolfram.com/ConvectiveOperator.html</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86&action=edit&section=11" title="節を編集: 関連項目"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E3%83%8A%E3%83%93%E3%82%A8-%E3%82%B9%E3%83%88%E3%83%BC%E3%82%AF%E3%82%B9%E3%81%AE%E5%BC%8F" class="mw-redirect" title="ナビエ-ストークスの式">ナビエ-ストークスの式</a></li> <li><a href="/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E6%96%B9%E7%A8%8B%E5%BC%8F_(%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6)" title="オイラー方程式 (流体力学)">オイラー方程式 (流体力学)</a></li> <li><a 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