Loading... !!! <!doctype html> <html> <head> <meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'> <link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; } .mac-os-11 { --title-bar-height:28px; } html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; } body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; } iframe { margin: auto; } a.url { word-break: break-all; } a:active, a:hover { outline: 0px; } .in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); } #write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; } #write.first-line-indent p { text-indent: 2em; } #write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; } #write.first-line-indent li { margin-left: 2em; } .for-image #write { padding-left: 8px; padding-right: 8px; } body.typora-export { padding-left: 30px; padding-right: 30px; } .typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; } .typora-export .task-list-item input { pointer-events: none; } @media screen and (max-width: 500px) { body.typora-export { padding-left: 0px; padding-right: 0px; } #write { padding-left: 20px; padding-right: 20px; } .CodeMirror-sizer { margin-left: 0px !important; } .CodeMirror-gutters { display: none !important; } } #write li > figure:last-child { margin-bottom: 0.5rem; } #write ol, #write ul { position: relative; } img { max-width: 100%; vertical-align: middle; image-orientation: from-image; } button, input, select, textarea { color: inherit; font: inherit; } input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; } *, ::after, ::before { box-sizing: border-box; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; } p { line-height: inherit; } h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; } p { orphans: 4; } h1 { font-size: 2rem; } h2 { font-size: 1.8rem; } h3 { font-size: 1.6rem; } h4 { font-size: 1.4rem; } h5 { font-size: 1.2rem; } h6 { font-size: 1rem; } .md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; } .hidden { display: none; } .md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; } a { cursor: pointer; } sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; } sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; } #write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; } figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; } figure > table { margin: 0px; } tr { break-inside: avoid; break-after: auto; } thead { display: table-header-group; } table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; } table.md-table td { min-width: 32px; } .CodeMirror-gutters { border-right: 0px; background-color: inherit; } .CodeMirror-linenumber { user-select: none; } .CodeMirror { text-align: left; } .CodeMirror-placeholder { opacity: 0.3; } .CodeMirror pre { padding: 0px 4px; } .CodeMirror-lines { padding: 0px; } div.hr:focus { cursor: none; } #write pre { white-space: pre-wrap; } #write.fences-no-line-wrapping pre { white-space: pre; } #write pre.ty-contain-cm { white-space: normal; } .CodeMirror-gutters { margin-right: 4px; } .md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; } .md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; } #write .md-fences.mock-cm { white-space: pre-wrap; } .md-fences.md-fences-with-lineno { padding-left: 0px; } #write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; } .md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; } .CodeMirror-line, twitterwidget { break-inside: avoid; } .footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; } .footnotes + .footnotes { margin-top: 0px; } .md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; } li div { padding-top: 0px; } blockquote { margin: 1rem 0px; } li .mathjax-block, li p { margin: 0.5rem 0px; } li blockquote { margin: 1rem 0px; } li { margin: 0px; position: relative; } blockquote > :last-child { margin-bottom: 0px; } blockquote > :first-child, li > :first-child { margin-top: 0px; } .footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; } #write .footnote-line { white-space: pre-wrap; } @media print { body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; } #write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; } .typora-export * { -webkit-print-color-adjust: exact; } .typora-export #write { break-after: avoid; } .typora-export #write::after { height: 0px; } .is-mac table { break-inside: avoid; } .typora-export-show-outline .typora-export-sidebar { display: none; } } .footnote-line { margin-top: 0.714em; font-size: 0.7em; } a img, img a { cursor: pointer; } pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; } p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; } #write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; } p > .md-image:only-child { display: inline-block; width: 100%; } #write .MathJax_Display { margin: 0.8em 0px 0px; } .md-math-block { width: 100%; } .md-math-block:not(:empty)::after { display: none; } .MathJax_ref { fill: currentcolor; } [contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; } .md-task-list-item { position: relative; list-style-type: none; } .task-list-item.md-task-list-item { padding-left: 0px; } .md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; } .math { font-size: 1rem; } .md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; } .md-toc-content { position: relative; margin-left: 0px; } .md-toc-content::after, .md-toc::after { display: none; } .md-toc-item { display: block; color: rgb(65, 131, 196); } .md-toc-item a { text-decoration: none; } .md-toc-inner:hover { text-decoration: underline; } .md-toc-inner { display: inline-block; cursor: pointer; } .md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; } .md-toc-h2 .md-toc-inner { margin-left: 2em; } .md-toc-h3 .md-toc-inner { margin-left: 4em; } .md-toc-h4 .md-toc-inner { margin-left: 6em; } .md-toc-h5 .md-toc-inner { margin-left: 8em; } .md-toc-h6 .md-toc-inner { margin-left: 10em; } @media screen and (max-width: 48em) { .md-toc-h3 .md-toc-inner { margin-left: 3.5em; } .md-toc-h4 .md-toc-inner { margin-left: 5em; } .md-toc-h5 .md-toc-inner { margin-left: 6.5em; } .md-toc-h6 .md-toc-inner { margin-left: 8em; } } a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; } .footnote-line a:not(.reversefootnote) { color: inherit; } .md-attr { display: none; } .md-fn-count::after { content: "."; } code, pre, samp, tt { font-family: var(--monospace); } kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; } .md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); } code { text-align: left; vertical-align: initial; } a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; } .os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; } .md-diagram-panel > svg { max-width: 100%; } [lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; } [lang="mermaid"] .node text { font-size: 1rem; } table tr th { border-bottom: 0px; } video { max-width: 100%; display: block; margin: 0px auto; } iframe { max-width: 100%; width: 100%; border: none; } .highlight td, .highlight tr { border: 0px; } mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); } .md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; } .md-expand mark .md-meta { opacity: 0.3 !important; } mark .md-meta { color: rgb(0, 0, 0); } @media print { .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; } } .md-diagram-panel .messageText { stroke: none !important; } .md-diagram-panel .start-state { fill: var(--node-fill); } .md-diagram-panel .edgeLabel rect { opacity: 1 !important; } .md-fences.md-fences-math { font-size: 1em; } .md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; } .md-fences-advanced:not(.md-focus) { background: inherit; } .typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; } .typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; } .typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; } .typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; } @media screen and (max-width: 1024px) { .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; } } @media screen and (max-width: 800px) { .typora-export-sidebar { display: none; } } .outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; } .outline-content ul { margin-top: 0px; margin-bottom: 0px; } .outline-content strong { font-weight: 400; } .outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; } .outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; } .outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; } .outline-expander:hover::before { content: ""; } .outline-h1 > .outline-item { padding-left: 0px; } .outline-h2 > .outline-item { padding-left: 1em; } .outline-h3 > .outline-item { padding-left: 2em; } .outline-h4 > .outline-item { padding-left: 3em; } .outline-h5 > .outline-item { padding-left: 4em; } .outline-h6 > .outline-item { padding-left: 5em; } .outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; } .outline-label:hover { text-decoration: underline; } .outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); } .outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; } .outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; } .outline-item-open > .outline-item > .outline-expander::before { content: ""; } .outline-children { display: none; } .info-panel-tab-wrapper { display: none; } .outline-item-open > .outline-children { display: block; } .typora-export .outline-item { padding-top: 1px; padding-bottom: 1px; } .typora-export .outline-item:hover { margin-right: -8px; border-right: 8px solid transparent; } .typora-export .outline-expander::before { content: "+"; font-family: inherit; top: -1px; } .typora-export .outline-expander:hover::before, .typora-export .outline-item-open > .outline-item > .outline-expander::before { content: "−"; } .typora-export-collapse-outline .outline-children { display: none; } .typora-export-collapse-outline .outline-item-open > .outline-children, .typora-export-no-collapse-outline .outline-children { display: block; } .typora-export-no-collapse-outline .outline-expander::before { content: "" !important; } .typora-export-show-outline .outline-item-active > .outline-item .outline-label { font-weight: 700; } .md-inline-math-container mjx-container { zoom: 0.95; } :root { --side-bar-bg-color: #fafafa; --control-text-color: #777; } @include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext); /* open-sans-regular - latin-ext_latin */ /* open-sans-italic - latin-ext_latin */ /* open-sans-700 - latin-ext_latin */ /* open-sans-700italic - latin-ext_latin */ html { font-size: 16px; -webkit-font-smoothing: antialiased; } body { font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif; color: rgb(51, 51, 51); line-height: 1.6; } #write { max-width: 860px; margin: 0 auto; padding: 30px; padding-bottom: 100px; } @media only screen and (min-width: 1400px) { #write { max-width: 1024px; } } @media only screen and (min-width: 1800px) { #write { max-width: 1200px; } } #write > ul:first-child, #write > ol:first-child{ margin-top: 30px; } a { color: #4183C4; } h1, h2, h3, h4, h5, h6 { position: relative; margin-top: 1rem; margin-bottom: 1rem; font-weight: bold; line-height: 1.4; cursor: text; } h1:hover a.anchor, h2:hover a.anchor, h3:hover a.anchor, h4:hover a.anchor, h5:hover a.anchor, h6:hover a.anchor { text-decoration: none; } h1 tt, h1 code { font-size: inherit; } h2 tt, h2 code { font-size: inherit; } h3 tt, h3 code { font-size: inherit; } h4 tt, h4 code { font-size: inherit; } h5 tt, h5 code { font-size: inherit; } h6 tt, h6 code { font-size: inherit; } h1 { font-size: 2.25em; line-height: 1.2; border-bottom: 1px solid #eee; } h2 { font-size: 1.75em; line-height: 1.225; border-bottom: 1px solid #eee; } /*@media print { .typora-export h1, .typora-export h2 { border-bottom: none; padding-bottom: initial; } .typora-export h1::after, .typora-export h2::after { content: ""; display: block; height: 100px; margin-top: -96px; border-top: 1px solid #eee; } }*/ h3 { font-size: 1.5em; line-height: 1.43; } h4 { font-size: 1.25em; } h5 { font-size: 1em; } h6 { font-size: 1em; color: #777; } p, blockquote, ul, ol, dl, table{ margin: 0.8em 0; } li>ol, li>ul { margin: 0 0; } hr { height: 2px; padding: 0; margin: 16px 0; background-color: #e7e7e7; border: 0 none; overflow: hidden; box-sizing: content-box; } li p.first { display: inline-block; } ul, ol { padding-left: 30px; } ul:first-child, ol:first-child { margin-top: 0; } ul:last-child, ol:last-child { margin-bottom: 0; } blockquote { border-left: 4px solid #dfe2e5; padding: 0 15px; color: #777777; } blockquote blockquote { padding-right: 0; } table { padding: 0; word-break: initial; } table tr { border: 1px solid #dfe2e5; margin: 0; padding: 0; } table tr:nth-child(2n), thead { background-color: #f8f8f8; } table th { font-weight: bold; border: 1px solid #dfe2e5; border-bottom: 0; margin: 0; padding: 6px 13px; } table td { border: 1px solid #dfe2e5; margin: 0; padding: 6px 13px; } table th:first-child, table td:first-child { margin-top: 0; } table th:last-child, table td:last-child { margin-bottom: 0; } .CodeMirror-lines { padding-left: 4px; } .code-tooltip { box-shadow: 0 1px 1px 0 rgba(0,28,36,.3); border-top: 1px solid #eef2f2; } .md-fences, code, tt { border: 1px solid #e7eaed; background-color: #f8f8f8; border-radius: 3px; padding: 0; padding: 2px 4px 0px 4px; font-size: 0.9em; } code { background-color: #f3f4f4; padding: 0 2px 0 2px; } .md-fences { margin-bottom: 15px; margin-top: 15px; padding-top: 8px; padding-bottom: 6px; } .md-task-list-item > input { margin-left: -1.3em; } @media print { html { font-size: 13px; } table, pre { page-break-inside: avoid; } pre { word-wrap: break-word; } } .md-fences { background-color: #f8f8f8; } #write pre.md-meta-block { padding: 1rem; font-size: 85%; line-height: 1.45; background-color: #f7f7f7; border: 0; border-radius: 3px; color: #777777; margin-top: 0 !important; } .mathjax-block>.code-tooltip { bottom: .375rem; } .md-mathjax-midline { background: #fafafa; } #write>h3.md-focus:before{ left: -1.5625rem; top: .375rem; } #write>h4.md-focus:before{ left: -1.5625rem; top: .285714286rem; } #write>h5.md-focus:before{ left: -1.5625rem; top: .285714286rem; } #write>h6.md-focus:before{ left: -1.5625rem; top: .285714286rem; } .md-image>.md-meta { /*border: 1px solid #ddd;*/ border-radius: 3px; padding: 2px 0px 0px 4px; font-size: 0.9em; color: inherit; } .md-tag { color: #a7a7a7; opacity: 1; } .md-toc { margin-top:20px; padding-bottom:20px; } .sidebar-tabs { border-bottom: none; } #typora-quick-open { border: 1px solid #ddd; background-color: #f8f8f8; } #typora-quick-open-item { background-color: #FAFAFA; border-color: #FEFEFE #e5e5e5 #e5e5e5 #eee; border-style: solid; border-width: 1px; } /** focus mode */ .on-focus-mode blockquote { border-left-color: rgba(85, 85, 85, 0.12); } header, .context-menu, .megamenu-content, footer{ font-family: "Segoe UI", "Arial", sans-serif; } .file-node-content:hover .file-node-icon, .file-node-content:hover .file-node-open-state{ visibility: visible; } .mac-seamless-mode #typora-sidebar { background-color: #fafafa; background-color: var(--side-bar-bg-color); } .md-lang { color: #b4654d; } /*.html-for-mac { --item-hover-bg-color: #E6F0FE; }*/ #md-notification .btn { border: 0; } .dropdown-menu .divider { border-color: #e5e5e5; opacity: 0.4; } .ty-preferences .window-content { background-color: #fafafa; } .ty-preferences .nav-group-item.active { color: white; background: #999; } .menu-item-container a.menu-style-btn { background-color: #f5f8fa; background-image: linear-gradient( 180deg , hsla(0, 0%, 100%, 0.8), hsla(0, 0%, 100%, 0)); } mjx-container[jax="SVG"] { direction: ltr; } mjx-container[jax="SVG"] > svg { overflow: visible; min-height: 1px; min-width: 1px; } mjx-container[jax="SVG"] > svg a { fill: blue; stroke: blue; } mjx-assistive-mml { position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; } mjx-assistive-mml[display="block"] { width: 100% !important; } mjx-container[jax="SVG"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="SVG"][display="true"][width="full"] { display: flex; } mjx-container[jax="SVG"][justify="left"] { text-align: left; } mjx-container[jax="SVG"][justify="right"] { text-align: right; } g[data-mml-node="merror"] > g { fill: red; stroke: red; } g[data-mml-node="merror"] > rect[data-background] { fill: yellow; stroke: none; } g[data-mml-node="mtable"] > line[data-line], svg[data-table] > g > line[data-line] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed { stroke-dasharray: 140; } g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted { stroke-linecap: round; stroke-dasharray: 0,140; } g[data-mml-node="mtable"] > g > svg { overflow: visible; } [jax="SVG"] mjx-tool { display: inline-block; position: relative; width: 0; height: 0; } [jax="SVG"] mjx-tool > mjx-tip { position: absolute; top: 0; left: 0; } mjx-tool > mjx-tip { display: inline-block; padding: .2em; border: 1px solid #888; font-size: 70%; background-color: #F8F8F8; color: black; box-shadow: 2px 2px 5px #AAAAAA; } g[data-mml-node="maction"][data-toggle] { cursor: pointer; } mjx-status { display: block; position: fixed; left: 1em; bottom: 1em; min-width: 25%; padding: .2em .4em; border: 1px solid #888; font-size: 90%; background-color: #F8F8F8; color: black; } foreignObject[data-mjx-xml] { font-family: initial; line-height: normal; overflow: visible; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 3; } g[data-mml-node="xypic"] path { stroke-width: inherit; } .MathJax g[data-mml-node="xypic"] path { stroke-width: inherit; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 0; } </style><title>曲线积分和曲面积分</title> </head> <body class='typora-export os-windows'><div class='typora-export-content'> <div id='write' class=''><h1 id='曲线积分和曲面积分'><span>曲线积分和曲面积分</span></h1><h2 id='引入'><span>引入</span></h2><p><span>为什么要曲线积分和曲面积分?</span></p><p><span>重积分是平面和空间内的区域上的积分</span></p><p><span>但是当我们需要对Jordan0测度的曲线上面的函数进行积分的时候,重积分无法做到</span></p><p><span>考虑数量场和向量场在空间内一个曲线和曲面的积分,我们需要引入</span><strong><span>曲线积分和曲面积分</span></strong></p><h2 id='数量场在曲线上的积分'><span>数量场在曲线上的积分</span></h2><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n8" cid="n8" mdtype="math_block" 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displaystyle="true"><mtr><mtd><mi>A</mi><mo>=</mo><msub><mi>M</mi><mn>0</mn></msub><mo>,</mo><msub><mi>M</mi><mn>1</mn></msub><mo>,</mo><msub><mi>M</mi><mn>2</mn></msub><mo>,</mo><mo>·</mo><mo>·</mo><mo>·</mo><mo>·</mo><mo>·</mo><mo>·</mo><msub><mi>M</mi><mi>n</mi></msub><mo>=</mo><mi>B</mi><mo>,</mo></mtd></mtr><mtr><mtd><mtext>其</mtext><mtext>中</mtext><mtext>分</mtext><mtext>割</mtext><mtext>点</mtext><mtext>的</mtext><mtext>坐</mtext><mtext>标</mtext><mtext>是</mtext><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>,</mo><msub><mi>z</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mtext>则</mtext></mtd></mtr><mtr><mtd><mi mathvariant="normal">Δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>=</mo><mover><mrow><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>M</mi><mi>i</mi></msub></mrow><mo>→</mo></mover><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mi>i</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>i</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>y</mi><mi>i</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>j</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>z</mi><mi>i</mi></msub><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><mtext>在</mtext><mtext>每</mtext><mtext>一</mtext><mtext>段</mtext><mtext>弧</mtext><mtext>上</mtext><mtext>任</mtext><mtext>取</mtext><mtext>一</mtext><mtext>点</mtext><msub><mi>N</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>η</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mtext>当</mtext><mtext>分</mtext><mtext>割</mtext><mtext>的</mtext><mtext>最</mtext><mtext>大</mtext><mtext>长</mtext><mtext>度</mtext><mtext>趋</mtext><mtext>于</mtext><mtext>零</mtext><mtext>时</mtext><mi>,</mi><mtext>如</mtext><mtext>果</mtext><mtext>下</mtext><mtext>列</mtext><mtext>和</mtext><mtext>式</mtext><mi>:</mi></mtd></mtr><mtr><mtd><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>v</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>η</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>·</mo><mi mathvariant="normal">Δ</mi><msub><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>η</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>η</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><msub><mi>y</mi><mi>i</mi></msub><mo>+</mo><mi>R</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>η</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><msub><mi>z</mi><mi>i</mi></msub><mtext>有</mtext><mtext>极</mtext><mtext>限</mtext></mtd></mtr><mtr><mtd><mtext>则</mtext><mtext>称</mtext><mtext>向</mtext><mtext>量</mtext><mtext>场</mtext><mtext>沿</mtext><mtext>着</mtext><mtext>曲</mtext><mtext>线</mtext><msub><mi>L</mi><mrow data-mjx-texclass="ORD"><mi>A</mi><mi>B</mi></mrow></msub><mtext>的</mtext><mtext>积</mtext><mtext>分</mtext><mi>,</mi><mtext>也</mtext><mtext>称</mtext><mtext>为</mtext><mtext>第</mtext><mtext>二</mtext><mtext>型</mtext><mtext>曲</mtext><mtext>线</mtext><mtext>积</mtext><mtext>分</mtext><mi>,</mi><mtext>记</mtext><mtext>为</mtext><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mrow data-mjx-texclass="ORD"><mi>A</mi><mi>B</mi></mrow></msub></mrow></msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mi>d</mi><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h3 id='计算-2'><span>计算</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n36" cid="n36" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" 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stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>j</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mi>z</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mtext>证</mtext><mtext>明</mtext><mi>:</mi><mtext>同</mtext><mtext>数</mtext><mtext>量</mtext><mtext>场</mtext><mtext>积</mtext><mtext>分</mtext><mi>(</mi><mtext>拉</mtext><mtext>格</mtext><mtext>朗</mtext><mtext>日</mtext><mtext>中</mtext><mtext>值</mtext><mtext>定</mtext><mtext>理</mtext><mi>)</mi></mtd></mtr><mtr><mtd><mtext>则</mtext><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>A</mi><mi>B</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mi>d</mi><mrow 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displaystyle="true"><mtr><mtd><mtext>证</mtext><mtext>明</mtext><mi>:</mi><msub><mo data-mjx-texclass="OP">∫</mo><mi>L</mi></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>1</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>2</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi></mtd></mtr><mtr><mtd><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>3</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>4</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi></mtd></mtr><mtr><mtd><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>1</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>3</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></msubsup><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></msubsup><mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mo>=</mo><mo>−</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mi>b</mi></msubsup><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>c</mi><mi>d</mi></msubsup><mfrac><mrow><mi>∂</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi><mo>=</mo><mo>−</mo><msub><mo data-mjx-texclass="OP">∫</mo><mi>D</mi></msub><mfrac><mrow><mi>∂</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>同</mtext><mtext>理</mtext><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>3</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>+</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mn>4</mn></msub></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∫</mo><mi>D</mi></msub><mfrac><mrow><mi>∂</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mtext>所</mtext><mtext>以</mtext><msub><mo data-mjx-texclass="OP">∫</mo><mi>L</mi></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∬</mo><mi>D</mi></msub><mo stretchy="false">[</mo><mfrac><mrow><mi>∂</mi><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>∂</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo stretchy="false">]</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div></blockquote><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n45" cid="n45" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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transform="translate(33207.4,0)"><use data-c="1D466" xlink:href="#MJX-594-TEX-I-1D466"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>得</mtext><mtext>大</mtext><mtext>区</mtext><mtext>间</mtext><mtext>的</mtext><mtext>边</mtext><mtext>界</mtext><mo>,</mo><mtext>两</mtext><mtext>个</mtext><mtext>方</mtext><mtext>块</mtext><mtext>的</mtext><mtext>重</mtext><mtext>叠</mtext><mtext>处</mtext><mtext>总</mtext><mtext>能</mtext><mtext>抵</mtext><mtext>消</mtext><msub><mo data-mjx-texclass="OP">∮</mo><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>D</mi></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∬</mo><mi>D</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><mi>∂</mi><mi>Q</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></math></mjx-assistive-mml></mjx-container></div></div><h3 id='推论116'><span>推论11.6 </span></h3><p><span>设D是满足Green 定理中条件的 区域,</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.925ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2177 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-128-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-128-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 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symmetric="true"></mo></mrow><mo>,</mo><mi>s</mi><mo>.</mo><mi>t</mi><mo>.</mo><mfrac><mrow><mi>∂</mi><mi>Q</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mtext>由</mtext><mi>G</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>n</mi><mtext>定</mtext><mtext>理</mtext><mi>,</mi><msub><mo data-mjx-texclass="OP">∮</mo><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>D</mi></mrow></msub><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∬</mo><mi>D</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><mi>∂</mi><mi>Q</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>∂</mi><mi>P</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mtd></mtr><mtr><mtd><msub><mo data-mjx-texclass="OP">∮</mo><mrow 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data-mml-node="mi" transform="translate(11923.1,0)"><use data-c="1D466" xlink:href="#MJX-598-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(12413.1,0)"><use data-c="29" xlink:href="#MJX-598-TEX-N-29"></use></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><mtext>证</mtext><mtext>明</mtext><mi>:</mi><mo>∵</mo><mtext>积</mtext><mtext>分</mtext><mtext>和</mtext><mtext>路</mtext><mtext>径</mtext><mtext>无</mtext><mtext>关</mtext><mo stretchy="false">⇒</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msub><mi>L</mi><mrow data-mjx-texclass="ORD"><mi>A</mi><mi>B</mi></mrow></msub></mrow></msub><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mi>d</mi><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>A</mi><mi>B</mi></msubsup><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mi>d</mi><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msubsup><mi>P</mi><mi>d</mi><mi>x</mi><mo>+</mo><mi>Q</mi><mi>d</mi><mi>y</mi><mo>=</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>∂</mi><mi>φ</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>P</mi></mtd></mtr><mtr><mtd><mtext>同</mtext><mtext>理</mtext><mfrac><mrow><mi>∂</mi><mi>φ</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mi>Q</mi><mtext>得</mtext><mtext>到</mtext><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold">∇</mi></mrow><mo data-mjx-texclass="NONE"></mo><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><p> </p><h2 id='数量场在曲面上的积分'><span>数量场在曲面上的积分</span></h2><p><img src="https://s2.loli.net/2022/04/28/EmXT9glIfRoS56r.png" referrerpolicy="no-referrer" alt="image-20220426092159167" style=""></p><p><span>假设三维空间中一个薄面S ,面密度为</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.17ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 517 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-131-TEX-I-1D70C" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g 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stretchy="false">→</mo></mover></mrow><mo>+</mo><mfrac><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mfrac><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mo stretchy="false">|</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><msup><mo stretchy="false">|</mo><mn>2</mn></msup><mo stretchy="false">|</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><msup><mo stretchy="false">|</mo><mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>⋅</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>v</mi><mo data-mjx-alternate="1">′</mo></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>E</mi><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi><mn>2</mn></msubsup><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>u</mi><mn>2</mn></msubsup></mtd></mtr><mtr><mtd><mi>F</mi><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>v</mi><mn>2</mn></msubsup><mo>=</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>v</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>v</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mrow data-mjx-texclass="ORD"><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>v</mi><mn>2</mn></msubsup></mtd></mtr><mtr><mtd><mi>G</mi><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi></msub><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>v</mi></msub><mo>=</mo><msubsup><mi>x</mi><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><msubsup><mi>x</mi><mi>v</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msubsup><mi>y</mi><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><msubsup><mi>y</mi><mi>v</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>u</mi><mo data-mjx-alternate="1">′</mo></msubsup><msubsup><mi>z</mi><mi>v</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd></mtr><mtr><mtd><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi></msub><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>v</mi></msub><mo>=</mo><msqrt><mi>E</mi><mi>G</mi><mo>−</mo><msup><mi>F</mi><mn>2</mn></msup></msqrt></mtd></mtr><mtr><mtd><mtext>记</mtext><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>=</mo><mfrac><mrow><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi></msub><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>v</mi></msub></mrow><mrow><mo stretchy="false">|</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>u</mi></msub><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mover><msup><mi>r</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mi>v</mi></msub><mo stretchy="false">|</mo></mrow></mfrac><mtext>为</mtext><mtext>单</mtext><mtext>位</mtext><mtext>法</mtext><mtext>向</mtext><mtext>量</mtext></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><h3 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stretchy="false">)</mo><msqrt><mi>E</mi><mi>G</mi><mo>−</mo><msup><mi>F</mi><mn>2</mn></msup></msqrt><mi>d</mi><mi>u</mi><mi>d</mi><mi>v</mi></mtd></mtr><mtr><mtd><mtext>特</mtext><mtext>别</mtext><mtext>地</mtext><mi>,</mi><mtext>当</mtext><mi>u</mi><mi>,</mi><mi>v</mi><mtext>是</mtext><mi>x</mi><mo>,</mo><mi>y</mi><mtext>时</mtext><mo>=</mo><msub><mo data-mjx-texclass="OP">∬</mo><mi>D</mi></msub><mi>φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msqrt><mn>1</mn><mo>+</mo><msubsup><mi>z</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi><mi>y</mi><mn>2</mn></msubsup></msqrt><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><h2 id='向量场在曲面上的积分'><span>向量场在曲面上的积分</span></h2><h3 id='曲面的定向'><span>曲面的定向</span></h3><p><img src="https://s2.loli.net/2022/04/28/FhSDHgx5X2AGUlI.png" referrerpolicy="no-referrer" alt="image-20220426095433629" style=""></p><p><strong><span>当且仅当曲面有连续的法向量场是曲面是可定向的</span></strong></p><h3 id='单侧曲面和双侧曲面'><span>单侧曲面和双侧曲面</span></h3><p><span>定义:</span></p><p><span>设S为光滑曲面(任何一点都有非零的法向量n(M)),取定一点</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.407ex" height="1.97ex" role="img" focusable="false" viewBox="0 -705 3274.1 870.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-132-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path id="MJX-132-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-132-TEX-N-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path id="MJX-132-TEX-I-1D446" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D440" xlink:href="#MJX-132-TEX-I-1D440"></use></g><g data-mml-node="mn" transform="translate(1003,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-132-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(1684.3,0)"><use data-c="2208" xlink:href="#MJX-132-TEX-N-2208"></use></g><g data-mml-node="mi" transform="translate(2629.1,0)"><use data-c="1D446" xlink:href="#MJX-132-TEX-I-1D446"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mn>0</mn></msub><mo>∈</mo><mi>S</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">M_0\in S</script></p><p><span>作任意一条S上过</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.182ex" height="1.92ex" role="img" focusable="false" viewBox="0 -683 1406.6 848.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.375ex;"><defs><path id="MJX-133-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 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404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-134-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-134-TEX-I-1D440" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path id="MJX-134-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-134-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D45B" xlink:href="#MJX-134-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(600,0)"><use data-c="28" xlink:href="#MJX-134-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(989,0)"><g data-mml-node="mi"><use data-c="1D440" xlink:href="#MJX-134-TEX-I-1D440"></use></g><g data-mml-node="mn" transform="translate(1003,-150) scale(0.707)"><use data-c="30" xlink:href="#MJX-134-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(2395.6,0)"><use data-c="29" xlink:href="#MJX-134-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo 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data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>θ</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>×</mo><msubsup><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo stretchy="false">→</mo></mover></mrow><mi>φ</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>∂</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>φ</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>s</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mi>θ</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>φ</mi><mo>,</mo><mi>s</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mi>θ</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>φ</mi><mo>,</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><p><span>2.对于</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="36.118ex" height="2.48ex" role="img" focusable="false" viewBox="0 -846 15964.2 1096" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-232-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 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