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  "contentMarkdown": "# 拉普拉斯方程到球谐函数\n> 创建时间：2021/6/1 14:55\n\n## 拉普拉斯方程到球谐函数\n\n  * 拉普拉斯方程到球谐函数\n    * Del算符与梯度、散度、旋度\n      * 梯度\n      * 散度\n      * 拉普拉斯算符（Laplacian）\n    * 拉普拉斯方程\n    * 球谐函数\n      * 球坐标系\n    * Ref\n\n## Del算符与梯度、散度、旋度\n\nDel算符 - 梯度、散度和旋度（Gradient, Divergence & Curl）是将Del算符应用于不同函数。\n\n详见：<https://zhuanlan.zhihu.com/p/349328782>\n\n### 梯度\n\n梯度的本意是一个向量（矢量），表示某一函数在该点处的方向导数沿着该方向取得最大值，即函数在该点处沿着该方向（此 **梯度的方向** ）变化最快，变化率最大（为该 **梯度的模** ）。\n\n![Alt text](assets/1622530604396.png)\n\n![Alt text](assets/1622531013847.png)\n\n![Alt text](assets/1622531070530.png)\n\n### 散度\n\n散度（divergence）可用于表征空间各点矢量场发散的强弱程度，物理上，散度的意义是场的有源性。当div F>0 ，表示该点有散发通量的正源（发散源）；当div F<0 表示该点有吸收通量的负源（洞或汇）；当div F=0，表示该点无源。\n\n在大气科学中散度指衡量速度场辐散、辐合强度的物理量。。三维空间的散度表示任意气块在单位时间内其单位体积的变化率。气块的体积膨胀称为辐散，气块体积收缩称为辐合。\n\n![Alt text](assets/1622531961145.png)\n\ndiv F 表示在某点处的单位体积内散发出来的矢量 F 的通量，所以 div F 描述了通量源的密度。\n\n散度是向量场的一种强度性质，就如同密度、浓度、温度一样，它对应的广延性质是一个封闭区域表面的通量。\n\n### 拉普拉斯算符（Laplacian）\n\nLaplacian算符 - 梯度的散度 Div(Grad(f))\n\n![Alt text](assets/1622532159801.png)\n\n或者\n\n![Alt text](assets/1622532164190.png)\n\n  * 告诉我们 **梯度（导数/斜率）的变化程度** ，可类比于单变量函数的二阶导数。\n\n  * 告诉我们 **一个点的函数值与周围环境其他点差异的均值** 。\n\n![Alt text](assets/1622532572788.png)\n\n  * 左侧谷底的梯度指向四周高的方向(向四周散开)，故梯度的散度为正，Laplacian>0；也可以理解为：谷底周围所有点都高于谷底，故高度差的均值(Laplacian)>0。\n\n  * 右侧谷峰的梯度指向中央的山峰(向山峰中央汇聚)，故梯度的散度为负，Laplacian<0；也可以理解为：山峰周围所有点都低于山峰，故高度差的均值(Laplacian)<0。\n\n  * 在山峰和谷底之间山坡的中点上，周围有的点高度比它高，有的比它低，所以故高度差的均值(Laplacian)=0。\n\n![Alt text](assets/1622532628358.png)\n\n## 拉普拉斯方程\n\n拉普拉斯方程（Laplace’s equation）又称调和方程、位势方程，是一种偏微分方程\n\n![Alt text](assets/1622532726645.png)\n\n<https://zhuanlan.zhihu.com/p/350388376>\nLaplacian等于0的函数被称为Harmonic，这种函数描述的场景是： **每个点的函数值等于它周围点的平均函数值** 。\n\n![Alt text](assets/1622532817997.png)\n\n拉普拉斯方程是关于不随时间变化的均衡状态(Equilibrium)的。在给定边界条件(Boundary Conditions)的情况下，在xy平面（或更高维）寻找x和y二阶导数方程等于0的解。\n\n## 球谐函数\n\n### 球坐标系\n\n![Alt text](assets/1622533943917.png)\n\n球坐标系(r,θ,φ)与直角坐标系(x,y,z)的转换关系:\nx=rsinθcosφ\ny=rsinθsinφ\nz=rcosθ\n\n## Ref\n\n<https://zhuanlan.zhihu.com/p/349328782>\n<https://baike.baidu.com/item/%E6%A2%AF%E5%BA%A6/13014729>\n<https://baike.baidu.com/item/%E7%90%83%E5%9D%90%E6%A0%87%E7%B3%BB>\n"
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