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target="_blank"><!--[--><!--[--><!--]--><!--]-->TinyEncrypt<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/ue5-style-guide" aria-label="UnrealStyleGuide" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->UnrealStyleGuide<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/AxTrace" aria-label="AxTrace" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->AxTrace<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/cyclone" aria-label="Cyclone" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->Cyclone<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading collapsible">学习笔记 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学相关 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="sph算法简介-一-数学基础" tabindex="-1"><a class="header-anchor" href="#sph算法简介-一-数学基础"><span>SPH算法简介（一）: 数学基础</span></a></h1><p>SPH(<a href="http://en.wikipedia.org/wiki/Smoothed_Particle_Hydrodynamics" target="_blank" rel="noopener noreferrer">Smoothed Particle Hydrodynamics</a>)算法是一种流体模拟算法，他的特点是简单快速，可以用在例如游戏这样的实时的交互软件中。SPH算法虽然简单，但要完全搞明白其中的原理和实现方法，也不是易事，写这个系列希望能全面介绍一下相关的内容，如果你搜索到这里，可以仔细看一下这个系列，希望能帮到你。<br> 烟雾、海浪、水滴…，这些司空见怪的自然现象其实有着非常复杂的数学规律，对于流体的研究，有两种完全不同的视角，分别是欧拉视角和拉格朗日视角。欧拉视角的坐标系是固定的，如同站在河边观察河水的流动一样，用这种视角分析流体需要建立网格单元，还会涉及到有限元等复杂的工程方法，一般用在离线的应用中。而拉格朗日视角则将流体视为流动的单元，例如将一片羽毛放入风中，那么羽毛的轨迹可以帮我们指示空气的流动规律。</p><figure><img src="/images/2014/08/sph_1.gif" alt="" tabindex="0" loading="lazy"><figcaption>欧拉视角 和 拉格朗日视角</figcaption></figure><p>SPH算法是典型的拉格朗日视角，它的基本原理就是通过粒子模拟流体的运动规律，然后再转换成网格进行流体渲染。</p><div align="center"><table align="center" class="invisibletable"><tbody><tr valign="center"><td><img src="/images/2014/08/sph_2.jpg"></td><td><p><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math></mjx-assistive-mml></mjx-container></p></td><td><img src="/images/2014/08/sph_3.jpg"></td><td><p><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math></mjx-assistive-mml></mjx-container></p></td><td><img src="/images/2014/08/sph_4.jpg"></td></tr></tbody></table></div><hr><p>在正式开始之前，需要把SPH算法涉及到的相关数学概念介绍一下，这些概念基本上都是大学数学中的内容，所以不用紧张，翻翻书就能想起来。</p><h4 id="标量场和矢量场" tabindex="-1"><a class="header-anchor" href="#标量场和矢量场"><span><strong>标量场和矢量场</strong></span></a></h4><p>如果空间区域内一点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container>，都有一个确定的数量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>M</mi></msub></math></mjx-assistive-mml></mjx-container>，则称这个空间区域内确定了一个标量场，如果空间区域内任意一点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container>，都有一个确定的向量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.735em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msub><mi>F</mi><mi>M</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，则称这空间区域内确定了一个矢量场。例如，液体中的密度，就是标量场，而速度，就是矢量场</p><h4 id="偏导数" tabindex="-1"><a class="header-anchor" href="#偏导数"><span><strong>偏导数</strong></span></a></h4><p>对于多元函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，定义<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></mjx-assistive-mml></mjx-container>在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><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></math></mjx-assistive-mml></mjx-container>处相对于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container>的偏导数为</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:22.645em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:22.645em;" side="right"><mjx-table style="width:auto;min-width:16.933em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-munder space="4"><mjx-row><mjx-base style="padding-left:0.352em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c25B3"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="3"><mjx-c class="mjx-c25B3"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c class="mjx-c25B3"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:22.645em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.249em;"><mjx-mtd id="mjx-eqn:1.1"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.249em;vertical-align:-0.789em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.1)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mfrac><mrow><mi>∂</mi><mi>z</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><munder><mo data-mjx-texclass="OP">lim</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">△</mi><mi>x</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi mathvariant="normal">△</mi><mi>x</mi><mo>,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><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><mi mathvariant="normal">△</mi><mi>x</mi></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>例如，定义<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup></math></mjx-assistive-mml></mjx-container>，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>∂</mi><mi>z</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>∂</mi><mi>x</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>,</mo><mi>∂</mi><mi>z</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>∂</mi><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container></p><h4 id="哈密顿算子" tabindex="-1"><a class="header-anchor" href="#哈密顿算子"><span><strong>哈密顿算子</strong></span></a></h4><p>哈密顿算子<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi></math></mjx-assistive-mml></mjx-container>在流体力学中是如此重要，以至于很多地方将这个符号作为流体力学的标志，所以这里要着重介绍一下，所谓“算子”，就是那种不能单独存在，必须和其他符号放在一起的一种数学符号，例如微分中的那个“<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></mjx-assistive-mml></mjx-container>”。哈密顿算子的定义如下：</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:16.395em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:16.395em;" side="right"><mjx-table style="width:auto;min-width:10.683em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.301em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.288em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:16.395em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.282em;"><mjx-mtd id="mjx-eqn:1.2"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.282em;vertical-align:-0.891em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.2)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∇</mi><mo>≡</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>z</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>z</mi></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>哈密顿算子有很多有趣的特性，它本身虽然并不是一个矢量，但很多运算确实可以把它视为一个矢量，例如把它作用在一个标量场<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>上，那么</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:17.29em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:17.29em;" side="right"><mjx-table style="width:auto;min-width:11.578em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.301em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.288em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:17.29em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.282em;"><mjx-mtd id="mjx-eqn:1.3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.282em;vertical-align:-0.891em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.3)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∇</mi><mi>A</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mrow><mi>∂</mi><mi>f</mi></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mrow><mi>∂</mi><mi>f</mi></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>z</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mrow><mi>∂</mi><mi>f</mi></mrow><mrow><mi>∂</mi><mi>z</mi></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>这个运算可以视为一个矢量和标量的乘法，得到的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container>是一个矢量场，称为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container>的“梯度”，顾名思义，梯度的含义就是标量场<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container>在某处变化快慢和方向，比如一个标量场<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>是一座高山在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>处的高度，则H的梯度是该高山在某处陡峭的程度，并且方向指向高处。</p><figure><img src="/images/2014/08/sph_8.gif" alt="" tabindex="0" loading="lazy"><figcaption>上面两个图中，标量场是黑白的，黑色表示大的数值，而其相应的梯度用蓝色箭头表示</figcaption></figure><p>而如果把哈密顿算子作用在一个矢量场<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.514em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>上，得到的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.514em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>称为矢量场<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container>的“散度”，散度的计算和矢量的点积运算相似，得到的是一个标量场。</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:28.391em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:28.391em;" side="right"><mjx-table style="width:auto;min-width:22.679em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mtable style="min-width:22.679em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.514em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c28 TEX-S3"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.301em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.288em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s3"><mjx-c class="mjx-c29 TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.314em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.301em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.288em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:28.391em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:5.061em;"><mjx-mtd id="mjx-eqn:1.4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:5.061em;vertical-align:-2.28em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.4)</mtext></mtd><mtd><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="normal">∇</mi><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>z</mi><mo stretchy="false">→</mo></mover></mrow><mfrac><mi>∂</mi><mrow><mi>∂</mi><mi>z</mi></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>⋅</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><msub><mi>A</mi><mi>x</mi></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>y</mi><mo stretchy="false">→</mo></mover></mrow><msub><mi>A</mi><mi>y</mi></msub><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi>z</mi><mo stretchy="false">→</mo></mover></mrow><msub><mi>A</mi><mi>z</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>∂</mi><msub><mi>A</mi><mi>x</mi></msub></mrow><mrow><mi>∂</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>∂</mi><msub><mi>A</mi><mi>y</mi></msub></mrow><mrow><mi>∂</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>∂</mi><msub><mi>A</mi><mi>z</mi></msub></mrow><mrow><mi>∂</mi><mi>z</mi></mrow></mfrac></mtd></mtr></mtable></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>散度的意义也很明显，就是描述一个矢量场“发散”的程度，例如下面的两个矢量场，左边的有很大的散度，而右边的散度为0</p><figure><img src="/images/2014/08/sph_10.gif" alt="" tabindex="0" loading="lazy"><figcaption>两个散度差别很大的矢量场</figcaption></figure><h4 id="拉普拉辛算子" tabindex="-1"><a class="header-anchor" href="#拉普拉辛算子"><span><strong>拉普拉辛算子</strong></span></a></h4><p>拉普拉辛算子<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup></math></mjx-assistive-mml></mjx-container>是二阶微分算子，有时也可写作<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c394"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi></math></mjx-assistive-mml></mjx-container>或者<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="3"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∇</mi><mo>⋅</mo><mi mathvariant="normal">∇</mi></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:16.615em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:16.615em;" side="right"><mjx-table style="width:auto;min-width:10.903em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:16.615em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.435em;"><mjx-mtd id="mjx-eqn:1.5"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.435em;vertical-align:-0.925em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.5)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mo>≡</mo><mfrac><msup><mi>∂</mi><mn>2</mn></msup><mrow><mi>∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><msup><mi>∂</mi><mn>2</mn></msup><mrow><mi>∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><msup><mi>∂</mi><mn>2</mn></msup><mrow><mi>∂</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>例如对于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:18.243em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:18.243em;" side="right"><mjx-table style="width:auto;min-width:12.531em;margin:0 2.856em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c2207"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.052em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D715"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:18.243em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.435em;"><mjx-mtd id="mjx-eqn:1.6"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c36"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.435em;vertical-align:-0.925em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(1.6)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>A</mi><mo>=</mo><mfrac><mrow><msup><mi>∂</mi><mn>2</mn></msup><mi>A</mi></mrow><mrow><mi>∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>∂</mi><mn>2</mn></msup><mi>A</mi></mrow><mrow><mi>∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>∂</mi><mn>2</mn></msup><mi>A</mi></mrow><mrow><mi>∂</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><!----><a class="route-link auto-link next" href="/blog/2025/02/SPH002.html" aria-label="Part2"><!--[--><div class="hint">Next <span class="arrow right"></span></div><div class="link"><span>Part2</span></div><!--]--></a></nav><!--[--><!--]--></main><!--]--></div><!--[--><!----><!--]--><!--]--></div>
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