AM@向量代数@方向导数与梯度

文章目录

• 概念
• 方向导数

• 例
• 方向导数公式

• 梯度

• 法向量
• 梯度的特点
• 梯度运算法则

概念

• 既有大小(模)又有方向的量,称为向量(或矢量)

• 印刷体常用黑体字母表示向量
• 手写通常用头箭头表示向量

• 向量的大小也被称为模
• 只考虑方向和大小(而不考虑起点)的向量称为自由向量
• 向量的坐标(表示):

• $a=(a_x,a_y,a_z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,简记为$a=(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 向量的模:$a=(a_x,a_y,a_z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则$|a|=\sqrt{a_x^2+a_y^2+a_z^2}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $a^2=a_x^2+a_y^2+a_z^2?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 零向量:模为0的向量称为零向量,其方向可以看作任意的,记为$\bold{0}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$或$\vec{0}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 单位向量:模为1的向量称为单位向量,通常向量$a?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的同向单位向量记为$a^{0}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 向量的坐标和单位向量表示加法表示

• 取$i=(1,0,0),j=(0,1,0),k=(0,0,1)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,它们分别是x,y,z轴的方向单位向量
• 则$a=(a_x,a_y,a_z)=a_xi+a_yj+a_zk,简记为a=xi+yj+zk?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 向量夹角

• 设向量$a=\overrightarrow{OA},b=\overrightarrow{OB}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则他们的夹角记为$\theta=\angle{AOB}=<a,b>,且\theta\in[0,\pi]?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 若$\theta=0?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则$a,b?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$通向
• 若$\theta=\pi?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则$a,b?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$反向
• 两者统称为$a,b?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$平行

• 向量的数量积(点积,内积)🎈

• 几何表示:$a\cdot{b}=|a||b|\cos{\theta},其中\theta={<a,b>}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 代数表示:$a=(a_x,a_y,a_z),b=(b_x,b_y,b_z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则$a\cdot{b}=a_xb_x+a_yb_y+a_zb_z?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 容易看出点积满足交换律$a\cdot{b}=b\cdot{a}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 特别的

• $b=a?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$时,$a^2=|a|^2=a_x^2+a_y^2+a_z^2?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 当b为单位向量时:$a\cdot{b}=|a||b|\cos{\theta}=|a|\cos{\theta},(|b|=1)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 数量乘的应用

• 求向量的模$|a|=\sqrt{a\cdot{a}}=\sqrt{a_x^2+a_y^2+a_z^2}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 求两个向量a,b的夹角余弦:$\cos{\theta}=\frac{a\cdot{b}}{|a||b|}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 判定两个向量垂直$a\perp{b}\Leftrightarrow{a\cdot{b}=0}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 空间向量方向余弦🎈

• 设向量$a\neq{0}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• $a?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$和$x,y,z?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$轴的正方向的夹角分别为$\alpha,\beta,\gamma?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,则称$\cos{\alpha},\cos{\beta},\cos{\gamma}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$为向量$a?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向余弦
• 一个向量的方向由方向余弦决定

• 向量在坐标轴上的投影🎈

• 设向量$a=(a_x,a_{y},a_z)=(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $\cos{\alpha}=\frac{x}{|a|}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• $\cos{\beta}=\frac{y}{|a|}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• $\cos{\gamma}=\frac{z}{|a|}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• $a^2=x^2+y^2+z^2?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $a=(x,y,z)=(\cos{\alpha}|a|,\cos{\beta}|a|,\cos{\gamma}|a|)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 容易看出$\cos^2{\alpha}+\cos^2{\beta}+\cos^2{\gamma}=1?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 因此,$a?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$ 的单位方向向量$a^{0}=(\cos{\alpha},\cos{\beta},\cos{\gamma})?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

方向导数

• 偏导数反映的是函数(自变量)沿着坐标轴方向的变换率
• 为例研究多元函数在某一点P沿任意方向(某个方向)的变化率,引入多元函数的方向导数的概念

• 例如,设$f(P)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$表示某物体内点P的温度,那么这个物体的热传导就依赖于温度沿某些方向的变化率

• 以三变量函数$f(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$为例

• 设$P(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$是给定点
• $l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$是从P出发的射线,它的方向向量用$\large{l}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$表示
• 设$P$是射线$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$上的任意一点

• $P$

• 其中,$(\Delta{x},\Delta{y},\Delta{z})=(\cos{\alpha}|a|,\cos{\beta}|a|,\cos{\gamma}|a|)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$🎈

• 记$|a|=|\overrightarrow{PP$表示$|PP$的长度
• 则$P$

• $\cos{\alpha},\cos{\beta},\cos{\gamma}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$s是向量$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向余弦
• 在$PP$这段长度内,函数$f?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的平均变化率为:

• $\frac{\Delta{f}}{|PP$

• 若该极限存在,则称其为$f?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$在点P沿方向$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向导数,记为

• $\frac{\partial{f(P)}}{\partial{l}}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 由于$P=(P_x,P_y,P_z)=(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,$\frac{\partial{f(P)}}{\partial{l}}=\frac{\partial{f(P_x,P_y,P_z)}}{\partial{l}}=\frac{\partial{f(x,y,z)}}{\partial{l}}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

例

• 设多元一次函数$f(x,y,z)=ax+by+cz?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$,向量$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向余弦为$\cos{\alpha},\cos{\beta},\cos{\gamma}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $f(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$沿$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$方向的平均变化率为

• $\frac{\Delta{f}}{|PP$
• 这表明,一次函数沿$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$方向的方向导数不随点的位置而改变

• 但是沿不同方向的方向导数一般不同

方向导数公式

• 如果$f(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$在点$P_0(x_0,y_0,z_0)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$可微,则

• $f(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$在$P_0?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$点沿任意方向$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向导数都存在,
• 且

• $\begin{aligned} \frac{\partial{f}}{\partial{l}} =&\frac{\partial{f}}{\partial{x}}(x_0,y_0,z_0)\cos{\alpha} +\frac{\partial{f}}{\partial{y}}(x_0,y_0,z_0)\cos{\beta} +\frac{\partial{f}}{\partial{z}}(x_0,y_0,z_0)\cos{\gamma} \\ =&\left.\frac{\partial{u}}{\partial{x}}\right|_{P_0}\cos{\alpha} +\left.\frac{\partial{u}}{\partial{y}}\right|_{P_0}\cos{\beta} +\left.\frac{\partial{u}}{\partial{z}}\right|_{P_0}\cos{\gamma} \\ &或另一种形式:\\ \frac{\partial{f}}{\partial{l}} =&f_x(x_0,y_0,z_0)\cos{\alpha} +f_y(x_0,y_0,z_0)\cos{\beta} +f_z(x_0,y_0,z_0)\cos{\gamma} \\=&f_x(P_0)\cos{\alpha} +f_y(P_0)\cos{\beta} +f_z(P_0)\cos{\gamma} \end{aligned} ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 证明:

• 设$P$是$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$上的点,则$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向余弦可以表示为:

• $\cos{\alpha}=\frac{\Delta{x}}{|PP$
• $\cos{\beta}=\frac{\Delta{y}}{|PP$
• $\cos{\gamma}=\frac{\Delta{z}}{|PP$
• $|PP$

• 由假设的$f(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$可微,由可微的定义:

• $\begin{aligned} f(P$
• 对两边同时除以$|PP$

• $\frac{f(P$

• 对两边取极限:

• $\begin{aligned} \frac{\partial{f}}{\partial{l}} =&\lim_{P$

梯度

• 梯度是一个与方向导数相关的概念
• 在研究一个物理量$u(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$在某一区域的分布时,常常需要考虑这个区域内由相同物理量的点,也就是使$u(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$取得相同值得各个点

• $u(x,y,z)=C ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 其中C使常数
• 该方程在几何上表示曲面(称为等量面)

• 例如气象学中的等压面
• 电学中的等位面

• 对于含有两个自变量的的物理量则有等量线

• 例如地图上的等高线可以反映地面的高低起伏
• 气象等温线表示地面气温变化

• 从等量面和等量线出发,引出向量函数

• 设$u=u(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$是一个数量函数,C是一个常数,关于u,C的等量面:

• $u(x,y,z)=C ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 设$P?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$是等量面上的任意一点,它的法向量为

• $F=\left.\frac{\partial{u}}{\partial{x}}\right|_{P}i +\left.\frac{\partial{u}}{\partial{y}}\right|_{P}j +\left.\frac{\partial{u}}{\partial{z}}\right|_{P}k ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 3个系数分别是u的偏导数在点$P?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的值

• 称向量$F?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$为数量函数$u?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$在$P?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的梯度(向量),记为

• $\bold{grad}{u}=\frac{\partial{u}}{\partial{x}}i+\frac{\partial{u}}{\partial{y}}j+\frac{\partial{u}}{\partial{x}}k?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $=(\frac{\partial{u}}{\partial{x}},\frac{\partial{u}}{\partial{y}},\frac{\partial{u}}{\partial{z}})?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 梯度长度:

• $|\bold{grad}{u}|=\sqrt{(\frac{\partial{u}}{\partial{x}})^2 +(\frac{\partial{u}}{\partial{y}})^2 +(\frac{\partial{u}}{\partial{z}})^2} ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

法向量

• 法向量，是空间解析几何的一个概念，垂直于平面的直线所表示的向量为该平面的法向量。
• 法向量适用于解析几何。由于空间内有无数个直线垂直于已知平面，因此一个平面都存在无数个法向量（包括两个单位法向量）。
• 三维平面的法线是垂直于该平面的三维向量。
• 曲面在某点P处的法线为垂直于该点切平面（tangent plane）的向量.

梯度的特点

• 梯度向量的方向是函数增长最快的方向,记为方向$d_m?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 梯度向量的模就是函数沿$d_m?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的变化率
• 证明:

• 设方向$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向余弦为$\cos{\alpha},\cos{\beta},\cos{\gamma}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 则$u(x,y,z)?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$沿$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向导数

• $\frac{\partial{u}}{\partial{l}} =\frac{\partial{u}}{\partial{x}}\cos{\alpha} +\frac{\partial{u}}{\partial{y}}\cos{\beta} +\frac{\partial{u}}{\partial{z}}\cos{\gamma} ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 令$l_0?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$是$l?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$的方向单位向量,则$l_0=(\cos{\alpha},\cos{\beta},\cos{\gamma})?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 则

• $\frac{\partial{u}}{\partial{l}} =(\frac{\partial{u}}{\partial{x}} ,\frac{\partial{u}}{\partial{y}} ,\frac{\partial{u}}{\partial{z}}) (\cos{\alpha},\cos{\beta},\cos{\gamma}) \\ =\bold{grad}u\cdot{l_0} =|\bold{grad}u|\cos{<\bold{grad}u,l_0>},(|l_0|=1) \\ 由:\frac{\partial{u}}{\partial{l}}=|\bold{grad}u|\cos{<\bold{grad}u,l_0>} \\记\theta=<\bold{grad}u,l_0> ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• 可以看出,当$\theta=0时?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$(也就是梯度向量与方向单位向量$l_0?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$同向),$\frac{\partial{u}}{\partial{l}}?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$取得最大值(方向导数取得最大值),

• 最大值为$|\bold{grad}u|?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• 也就函数$u?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$沿着梯度向量变化时变化最快,变化率是最大变化率

梯度运算法则

• $\bold{grad}(u_1\pm{u_2})=\bold{grad}u_1\pm\bold{grad}u_2?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$
• $\bold{grad}u_1u_2=u_1\bold{grad}u_2+u_2\bold{grad}u_1?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $\bold{grad}{u}=\frac{\partial{u}}{\partial{x}}i+\frac{\partial{u}}{\partial{y}}j+\frac{\partial{u}}{\partial{x}}k \\ 令u=u_1u_2 \\ \frac{\partial{u_1u_2}}{\partial{x}}i =(\frac{\partial{u_1}}{\partial{x}}u_2 +u_1\frac{\partial{u_2}}{\partial{x}})i \\ \frac{\partial{u_1u_2}}{\partial{y}}j =(\frac{\partial{u_1}}{\partial{y}}u_2 +u_1\frac{\partial{u_2}}{\partial{y}})j \\ \frac{\partial{u_1u_2}}{\partial{z}}k =(\frac{\partial{u_1}}{\partial{z}}u_2 +u_1\frac{\partial{u_2}}{\partial{z}})k \\ 上述3个式子两侧分别相加: \\ \bold{grad}{u_1u_2}= u_2(\frac{\partial{u_1}}{\partial{x}}i+\frac{\partial{u_1}}{\partial{y}}j+\frac{\partial{u_1}}{\partial{z}}k) +u_1(\frac{\partial{u_2}}{\partial{x}}i+\frac{\partial{u_2}}{\partial{y}}j+\frac{\partial{u_2}}{\partial{z}}k) \\ =u_1\bold{grad}u_2+u_2\bold{grad}u_1 ?x-oss-process=image/watermark,size_16,text_QDUxQ1RP5Y2a5a6i,color_FFFFFF,t_30,g_se,x_10,y_10,shadow_20,type_ZmFuZ3poZW5naGVpdGk=$

• $\bold{grad}F(u)=F$

• $\bold{grad}F(u)= \frac{\partial{F(u)}}{\partial{x}}i+\frac{\partial{F(u)}}{\partial{y}}j+\frac{\partial{F(u)}}{\partial{x}}k\\ =\frac{\partial{F(u)}}{\partial{u}}\frac{\partial{u}}{\partial{x}}i +\frac{\partial{F(u)}}{\partial{u}}\frac{\partial{u}}{\partial{y}}j +\frac{\partial{F(u)}}{\partial{u}}\frac{\partial{u}}{\partial{z}}k \\ =\frac{\partial{F(u)}}{\partial{u}}(\frac{\partial{u}}{\partial{x}}i+\frac{\partial{u}}{\partial{y}}j+\frac{\partial{u}}{\partial{x}}k) \\ =\frac{\partial{F(u)}}{\partial{u}}\bold{grad}u \\ 或 =F$