梯度@等值线@梯度运算法则
文章目录
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- 梯度
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- 点处梯度
- 函数梯度
- 梯度和方向导数的关系
- 等值线
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- 等值线法线和梯度
- 三元函数梯度
- 点处梯度
- 函数梯度
- 梯度长度
- 等值面
- 梯度运算法则
梯度
- 梯度是一个与方向导数相关的概念,梯度本质上是向量,是由各个自变量的偏导数定义的向量;梯度通常充当方向导数(函数变化率)的最值的角色
点处梯度
- 在二元函数的情形下,设函数 f ( x , y ) f(x,y) f(x,y)在平面区域 D D D内具有一阶连续偏导数,则对于没一点 P 0 ( x 0 , y 0 ) ∈ D P_{0}(x_0,y_0)\in{D} P0(x0,y0)∈D,都可以定出一个向量,其坐标分解式为: f x ( x 0 , y 0 ) i f_{x}(x_0,y_0)\bold{i} fx(x0,y0)i+ f y ( x 0 + y 0 ) j f_{y}(x_0+y_0)\bold{j} fy(x0+y0)j
- 该向量称为函数 f ( x , y ) f(x,y) f(x,y)在点 P 0 ( x 0 , y 0 ) P_{0}(x_0,y_0) P0(x0,y0)的梯度(或梯度向量),记为 g r a d f ( x 0 , y 0 ) \bold{grad}{f(x_0,y_0)} gradf(x0,y0)或 ∇ f ( x 0 , y 0 ) \nabla{f(x_0,y_0)} ∇f(x0,y0),即:
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g r a d f ( x 0 , y 0 ) \bold{grad}{f(x_0,y_0)} gradf(x0,y0)= ∇ f ( x 0 , y 0 ) \nabla{f(x_0,y_0)} ∇f(x0,y0)= f x ( x 0 , y 0 ) i + f y ( x 0 , y 0 ) j f_{x}(x_0,y_0)\bold{i}+f_{y}(x_0,y_0)\bold{j} fx(x0,y0)i+fy(x0,y0)j;若向量写成坐标式,为 g r a d f ( x 0 , y 0 ) \bold{grad}{f(x_0,y_0)} gradf(x0,y0)= ( f x ( x 0 , y 0 ) , f y ( x 0 , y 0 ) ) (f_{x}(x_0,y_0),f_{y}(x_0,y_0)) (fx(x0,y0),fy(x0,y0))
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其中 ∇ \nabla ∇= ∂ ∂ x i + ∂ ∂ y j \frac{\partial}{\partial{x}}\bold{i}+\frac{\partial}{\partial{y}}\bold{j} ∂x∂i+∂y∂j,称为二维向量微分算子或Nabla算子
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∇ f \nabla{f} ∇f= ∂ f ∂ x i + ∂ f ∂ x j \frac{\partial{f}}{\partial{x}}\bold{i}+\frac{\partial{f}}{\partial{x}}\bold{j} ∂x∂fi+∂x∂fj
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- 这种定义是抽象自方向导数的计算公式
函数梯度
- g r a d f ( x , y ) \bold{grad}{f(x,y)} gradf(x,y)= ( f x ( x , y ) , f y ( x , y ) ) (f_{x}(x,y),f_{y}(x,y)) (fx(x,y),fy(x,y))= ( z x , z y ) (z_{x},z_{y}) (zx,zy)
梯度和方向导数的关系
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若函数 f ( x , y ) f(x,y) f(x,y)在点 P 0 ( x 0 , y 0 ) P_0(x_0,y_0) P0(x0,y0)可微分, e l \bold{e}_{l} el= ( cos α , cos β ) (\cos\alpha,\cos\beta) (cosα,cosβ)是与方向 l l l同向的单位向量,则
- ∂ z ∂ l ∣ ( x 0 , y 0 ) \frac{\partial{z}}{\partial{l}}|_{(x_0,y_{0})} ∂l∂z∣(x0,y0)= f x ( x 0 , y 0 ) cos α + f y ( x 0 , y 0 ) cos β f_{x}(x_0,y_0)\cos{\alpha}+f_{y}(x_0,y_0)\cos{\beta} fx(x0,y0)cosα+fy(x0,y0)cosβ= ( f x ( x 0 , y 0 ) , f y ( x 0 , y 0 ) ) ( cos α , cos β ) (f_{x}(x_0,y_0),f_{y}(x_0,y_0))(\cos\alpha,\cos\beta) (fx(x0,y0),fy(x0,y0))(cosα,cosβ)
- = g r a d f ( x 0 , y 0 ) ⋅ e l \bold{grad}{f(x_0,y_0)}\cdot{\bold{e}_{l}} gradf(x0,y0)⋅el= ∣ g r a d f ( x 0 , y 0 ) ∣ ∣ e l ∣ ⋅ cos θ |\bold{grad}{f(x_0,y_0)}||\bold{e}_{l}|\cdot\cos\theta ∣gradf(x0,y0)∣∣el∣⋅cosθ
- = ∣ g r a d f ( x 0 , y 0 ) ∣ ⋅ cos θ |\bold{grad}{f(x_0,y_0)}|\cdot\cos\theta ∣gradf(x0,y0)∣⋅cosθ
(1)
- 其中 θ = < g r a d f ( x 0 , y 0 ) , e l > \theta=<\bold{grad}{f(x_0,y_0)},\bold{e}_{l}> θ=<gradf(x0,y0),el>
- ∂ z ∂ l ∣ ( x 0 , y 0 ) \frac{\partial{z}}{\partial{l}}|_{(x_0,y_{0})} ∂l∂z∣(x0,y0)= f x ( x 0 , y 0 ) cos α + f y ( x 0 , y 0 ) cos β f_{x}(x_0,y_0)\cos{\alpha}+f_{y}(x_0,y_0)\cos{\beta} fx(x0,y0)cosα+fy(x0,y0)cosβ= ( f x ( x 0 , y 0 ) , f y ( x 0 , y 0 ) ) ( cos α , cos β ) (f_{x}(x_0,y_0),f_{y}(x_0,y_0))(\cos\alpha,\cos\beta) (fx(x0,y0),fy(x0,y0))(cosα,cosβ)
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式(1)表明,函数在一点的梯度与函数在这点的方向导数间存在关系
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梯度向量的方向是函数增长最快的方向;梯度向量反向是函数减少最快的方向
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梯度向量的模就是函数沿梯度方向的变化率
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当 θ = 0 \theta=0 θ=0时,即方向 e l \bold{e}_{l} el与梯度 g r a d f ( x 0 , y 0 ) \bold{grad}f(x_0,y_0) gradf(x0,y0)的方向相同时,函数 f ( x , y ) f(x,y) f(x,y)增加最快
- 函数在梯度方向的方向导数达到最大值,这个最大值就是梯度的模,即 ∂ z ∂ l ∣ ( x 0 , y 0 ) \frac{\partial{z}}{\partial{l}}|_{(x_0,y_{0})} ∂l∂z∣(x0,y0)= ∣ g r a d f ( x 0 , y 0 ) ∣ |\bold{grad}f(x_0,y_0)| ∣gradf(x0,y0)∣
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当 θ = π \theta=\pi θ=π,即方向 e l \bold{e}_{l} el与梯度 g r a d f ( x 0 , y 0 ) \bold{grad}f(x_0,y_0) gradf(x0,y0)的方向相反时,函数 f ( x , y ) f(x,y) f(x,y)减少最快
- 函数在这个方向的方向导数达到最小值,即 ∂ z ∂ l ∣ ( x 0 , y 0 ) \frac{\partial{z}}{\partial{l}}|_{(x_0,y_{0})} ∂l∂z∣(x0,y0)= − ∣ g r a d f ( x 0 , y 0 ) ∣ -|\bold{grad}f(x_0,y_0)| −∣gradf(x0,y0)∣
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当 θ = π 2 \theta=\frac{\pi}{2} θ=2π,即方向 e l \bold{e}_{l} el与梯度 g r a d f ( x 0 , y 0 ) \bold{grad}f(x_0,y_0) gradf(x0,y0)的方向正交时,函数的变化率为0,即
- ∂ z ∂ l ∣ ( x 0 , y 0 ) \frac{\partial{z}}{\partial{l}}|_{(x_0,y_{0})} ∂l∂z∣(x0,y0)= ∣ g r a d f ( x 0 , y 0 ) ∣ cos π 2 |\bold{grad}f(x_0,y_0)|\cos\frac{\pi}{2} ∣gradf(x0,y0)∣cos2π=0
等值线
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在研究一个物理量 u ( x , y , z ) u(x,y,z) u(x,y,z)在某一区域的分布时,常常需要考虑这个区域内有相同物理量的点,也就是使 u ( x , y , z ) u(x,y,z) u(x,y,z)取得相同值得各个点
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一般地,二元函数 z = f ( x , y ) z=f(x,y) z=f(x,y)在几何上时一个曲面 C C C
- 若用一个平面 z = c z=c z=c,( c c c是常数)去截该曲面 C C C得的曲线 L L L的方程为 z = f ( x , y ) z=f(x,y) z=f(x,y); z = c z=c z=c
(1) - 则曲线 L L L在 x O y xOy xOy面上的投影式一条平面曲线 L ′ L' L′,其方程为 f ( x , y ) = c f(x,y)=c f(x,y)=c(将方程组(1)中的 z z z消去即得)
- 对于 L ′ L' L′上的一切点 ( x , y ) (x,y) (x,y), f f f的函数值都为 f ( x , y ) = c f(x,y)=c f(x,y)=c,因此称平面曲线 L ′ L' L′为函数 z = f ( x , y ) z=f(x,y) z=f(x,y)的等值线(等量线)
- 若用一个平面 z = c z=c z=c,( c c c是常数)去截该曲面 C C C得的曲线 L L L的方程为 z = f ( x , y ) z=f(x,y) z=f(x,y); z = c z=c z=c
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若 f x , f y f_x,f_y fx,fy不同时为0,则等值线 L ′ : f ( x , y ) = c L':f(x,y)=c L′:f(x,y)=c上任意一点 P 0 ( x 0 , y 0 ) P_{0}(x_0,y_0) P0(x0,y0)处的一个
- 法向量为 m = ( f x ( x 0 , y 0 ) , f y ( x 0 , y 0 ) ) \bold{m}=(f_{x}(x_0,y_0),f_{y}(x_0,y_0)) m=(fx(x0,y0),fy(x0,y0))
- 单位法向量为 n \bold{n} n= 1 f x 2 ( x 0 , y 0 ) + f y 2 ( x 0 , y 0 ) ( f x ( x 0 , y 0 ) , f y ( x 0 , y 0 ) ) \frac{1}{\sqrt{f_{x}^2(x_0,y_0)+f_{y}^{2}(x_0,y_0)}}(f_{x}(x_0,y_0),f_{y}(x_0,y_0)) fx2(x0,y0)+fy2(x0,y0)1(fx(x0,y0),fy(x0,y0))= g r a d f ( x 0 , y 0 ) ∣ g r a d f ( x 0 , y 0 ) ∣ \frac{\bold{grad}{f(x_0,y_0)}}{|\bold{grad}f(x_0,y_0)|} ∣gradf(x0,y0)∣gradf(x0,y0)
(2) - 将(2)变形,可得 g r a d f ( x 0 , y 0 ) \bold{grad}f(x_0,y_0) gradf(x0,y0)= ∣ g r a d f ( x 0 , y 0 ) ∣ ⋅ n |\bold{grad}{f(x_0,y_0)}|\cdot{\bold{n}} ∣gradf(x0,y0)∣⋅n
(2-1)
等值线法线和梯度
- 公式(2)表明函数 f ( x , y ) f(x,y) f(x,y)在点 P 0 ( x 0 , y 0 ) P_{0}(x_0,y_0) P0(x0,y0)的梯度 g r a d f ( x 0 , y 0 ) \bold{grad}{f(x_0,y_0)} gradf(x0,y0)的方向就是等值线 f ( x , y ) = c f(x,y)=c f(x,y)=c在 P 0 P_{0} P0点的法线方向 n \bold{n} n
- 对于二元函数 z = f ( x , y ) z=f(x,y) z=f(x,y)而言,其在 x O y xOy xOy上的投影等值线的法向量平行于 x O y xOy xOy,对应的法线属于 x O y xOy xOy
- 而梯度的模 ∣ g r a d f ( x 0 , y 0 ) ∣ |\bold{grad}{f(x_0,y_0)}| ∣gradf(x0,y0)∣就是沿法线方向(梯度方向)的方向导数 ∂ f ∂ n \frac{\partial{f}}{\partial{\bold{n}}} ∂n∂f,即 ∂ f ∂ n \frac{\partial{f}}{\partial{\bold{n}}} ∂n∂f= ∣ g r a d f ( x 0 , y 0 ) ∣ |\bold{grad}{f(x_0,y_0)}| ∣gradf(x0,y0)∣
(3) - 于是根据向量可以表示为该向量的模长乘以该向量的单位方向向量,有 g r a d f ( x 0 , y 0 ) \bold{grad}f(x_0,y_0) gradf(x0,y0)= ∂ f ∂ n n \frac{\partial{f}}{\partial{\bold{n}}}\bold{n} ∂n∂fn
(4),代入(3)可知,式(4)和(2-1)是相当的
三元函数梯度
点处梯度
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二元函数的梯度概念可以类似地推广到三元函数的情形
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设函数 u = f ( x , y , z ) u=f(x,y,z) u=f(x,y,z)在空间区域 G G G内具有一阶连续偏导数,则对于每一点 P 0 ( x 0 , y 0 , z 0 ) ∈ G P_{0}(x_0,y_0,z_0)\in{G} P0(x0,y0,z0)∈G,都可以定义处一个向量: ∂ u ∂ x ∣ P 0 i + ∂ u ∂ y ∣ P 0 j + ∂ u ∂ x ∣ P 0 k \frac{\partial{u}}{\partial{x}}|_{P_{0}}i+\frac{\partial{u}}{\partial{y}}|_{P_{0}}j+\frac{\partial{u}}{\partial{x}}|_{P_{0}}k ∂x∂u∣P0i+∂y∂u∣P0j+∂x∂u∣P0k,即 f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f x ( x 0 , y 0 , z 0 ) k f_{x}(x_0,y_0,z_0)\bold{i}+f_{y}(x_0,y_0,z_0)\bold{j}+f_{x}(x_0,y_0,z_0)\bold{k} fx(x0,y0,z0)i+fy(x0,y0,z0)j+fx(x0,y0,z0)k,此向量称为函数 f ( x , y , z ) f(x,y,z) f(x,y,z)在点 P 0 P_{0} P0处的梯度,记为: g r a d u ∣ P 0 \bold{grad}{u}|_{P_0} gradu∣P0或 g r a d f ( x 0 , y 0 , z 0 ) \bold{grad}{f(x_0,y_0,z_0)} gradf(x0,y0,z0)或 ∇ f ( x 0 , y 0 , z 0 ) \nabla{f(x_0,y_0,z_0)} ∇f(x0,y0,z0),即
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g r a d u ∣ P 0 \bold{grad}{u}|_{P_0} gradu∣P0= g r a d f ( x 0 , y 0 , z 0 ) \bold{grad}{f(x_0,y_0,z_0)} gradf(x0,y0,z0)= ∇ f ( x 0 , y 0 , z 0 ) \nabla{f(x_0,y_0,z_0)} ∇f(x0,y0,z0)= f x ( x 0 , y 0 , z 0 ) i + f y ( x 0 , y 0 , z 0 ) j + f x ( x 0 , y 0 , z 0 ) k f_{x}(x_0,y_0,z_0)\bold{i}+f_{y}(x_0,y_0,z_0)\bold{j}+f_{x}(x_0,y_0,z_0)\bold{k} fx(x0,y0,z0)i+fy(x0,y0,z0)j+fx(x0,y0,z0)k
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其中 ∇ \nabla ∇= ∂ ∂ x i + ∂ ∂ y j + ∂ ∂ z k \frac{\partial}{\partial{x}}\bold{i}+\frac{\partial}{\partial{y}}\bold{j}+\frac{\partial}{\partial{z}}\bold{k} ∂x∂i+∂y∂j+∂z∂k,称为三维向量微分算子或Nabla算子
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∇ f \nabla{f} ∇f= ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k \frac{\partial{f}}{\partial{x}}\bold{i}+\frac{\partial{f}}{\partial{y}}\bold{j}+\frac{\partial{f}}{\partial{z}}\bold{k} ∂x∂fi+∂y∂fj+∂z∂fk
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三元函数梯度和二元函数梯度有完全类似的结论
函数梯度
- g r a d f ( x , y , z ) \bold{grad}{f(x,y,z)} gradf(x,y,z)= ( f x ( x , y , z ) , f y ( x , y , z ) , f z ( x , y , z ) ) (f_{x}(x,y,z),f_{y}(x,y,z),f_{z}(x,y,z)) (fx(x,y,z),fy(x,y,z),fz(x,y,z))= ( u x , u y , u z ) (u_{x},u_{y},u_{z}) (ux,uy,uz)
梯度长度
- ∣ g r a d u ∣ = ( ∂ u ∂ x ) 2 + ( ∂ u ∂ y ) 2 + ( ∂ u ∂ z ) 2 |\bold{grad}{u}|=\sqrt{(\frac{\partial{u}}{\partial{x}})^2 +(\frac{\partial{u}}{\partial{y}})^2 +(\frac{\partial{u}}{\partial{z}})^2} ∣gradu∣=(∂x∂u)2+(∂y∂u)2+(∂z∂u)2
等值面
- 若引入去曲面: f ( x , y , z ) = c f(x,y,z)=c f(x,y,z)=c为函数 f ( x , y , z ) f(x,y,z) f(x,y,z)的等值面,可得 f ( x , y , z ) f(x,y,z) f(x,y,z)在一点 P 0 ( x 0 , y 0 , z 0 ) P_0(x_0,y_0,z_0) P0(x0,y0,z0)的梯度 ∇ f ( x 0 , y 0 , z 0 ) \nabla{f(x_0,y_0,z_0)} ∇f(x0,y0,z0)的方向就是等值面 f ( x , y , z ) = c f(x,y,z)=c f(x,y,z)=c在这点的法线方向 n \bold{n} n
- 法向量为 m = ( f x ( x 0 , y 0 , z 0 ) , f y ( x 0 , y 0 , z 0 ) , f z ( x 0 , y 0 , z 0 ) ) \bold{m}=(f_{x}(x_0,y_0,z_0),f_{y}(x_0,y_0,z_0),f_{z}(x_0,y_0,z_0)) m=(fx(x0,y0,z0),fy(x0,y0,z0),fz(x0,y0,z0)),恰为 g r a d f ( x 0 , y 0 , z 0 ) \bold{grad}{f(x_0,y_0,z_0)} gradf(x0,y0,z0)
- 单位法向量为 n \bold{n} n= 1 ∣ m ∣ m \frac{1}{|\bold{m}|}\bold{m} ∣m∣1m= g r a d f ( x 0 , y 0 , z 0 ) ∣ g r a d f ( x 0 , y 0 , z 0 ) ∣ \frac{\bold{grad}{f(x_0,y_0,z_0)}}{|\bold{grad}f(x_0,y_0,z_0)|} ∣gradf(x0,y0,z0)∣gradf(x0,y0,z0)
- 并且 ∂ f ∂ n \frac{\partial{f}}{\partial{\bold{n}}} ∂n∂f= ∣ g r a d f ( x 0 , y 0 , z 0 ) ∣ |\bold{grad}{f(x_0,y_0,z_0)}| ∣gradf(x0,y0,z0)∣,即 g r a d f ( x 0 , y 0 , z 0 ) \bold{grad}{f(x_0,y_0,z_0)} gradf(x0,y0,z0)= ∂ f ∂ n n \frac{\partial{f}}{\partial{\bold{n}}}\bold{n} ∂n∂fn
梯度运算法则
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g r a d ( u 1 ± u 2 ) \bold{grad}(u_1\pm{u_2}) grad(u1±u2)= g r a d u 1 ± g r a d u 2 \bold{grad}u_1\pm\bold{grad}u_2 gradu1±gradu2
- 令 u = u 1 + u 2 u=u_1+u_2 u=u1+u2;规定, u 1 x u_{1x} u1x表示对 u 1 u_1 u1求关于 x x x的偏导, u 2 x , u 1 y , u 2 y u_{2x},u_{1y},u_{2y} u2x,u1y,u2y并作类似的规定
- 等式 g r a d u \bold{grad}{u} gradu= ( u x , u y ) (u_{x},u_{y}) (ux,uy)= ( u 1 x + u 2 x , u 1 y + u 2 y ) (u_{1x}+u_{2x},u_{1y}+u_{2y}) (u1x+u2x,u1y+u2y)= ( u 1 x , u 1 y ) + ( u 2 x , u 2 y ) (u_{1x},u_{1y})+(u_{2x},u_{2y}) (u1x,u1y)+(u2x,u2y)= g r a d u 1 ± g r a d u 2 \bold{grad}u_1\pm\bold{grad}u_2 gradu1±gradu2
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g r a d u 1 u 2 \bold{grad}u_1u_2 gradu1u2= u 1 g r a d u 2 + u 2 g r a d u 1 u_1\bold{grad}u_2+u_2\bold{grad}u_1 u1gradu2+u2gradu1
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下面用2套符号分别推导 u 1 , u 2 u_1,u_2 u1,u2为二元和三元函数情形下的法则成立
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令 u = u 1 u 2 u=u_1u_2 u=u1u2
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二元情形:
- g r a d u \bold{grad}u gradu= ( u x , u y ) (u_{x},u_{y}) (ux,uy)= ( u 1 x u 2 + u 1 u 2 x , u 1 y u 2 + u 1 u 2 y ) (u_{1x}u_2+u_1u_{2x},u_{1y}u_{2}+u_{1}u_{2y}) (u1xu2+u1u2x,u1yu2+u1u2y)= ( u 1 x u 2 , u 1 y u 2 ) (u_{1x}u_2,u_{1y}u_2) (u1xu2,u1yu2)+ ( u 1 u 2 x , u 1 u 2 y ) (u_{1}u_{2x},u_1u_{2y}) (u1u2x,u1u2y)= u 2 ( u 1 x , u 1 y ) u_{2}(u_{1x},u_{1y}) u2(u1x,u1y)+ u 1 ( u 2 x , u 2 y ) u_{1}(u_{2x},u_{2y}) u1(u2x,u2y)= u 1 g r a d u 2 + u 2 g r a d u 1 u_1\bold{grad}u_2+u_2\bold{grad}u_1 u1gradu2+u2gradu1
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三元情形:
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g r a d u = ∂ u ∂ x i + ∂ u ∂ y j + ∂ u ∂ x k ∂ u 1 u 2 ∂ x i = ( ∂ u 1 ∂ x u 2 + u 1 ∂ u 2 ∂ x ) i ∂ u 1 u 2 ∂ y j = ( ∂ u 1 ∂ y u 2 + u 1 ∂ u 2 ∂ y ) j ∂ u 1 u 2 ∂ z k = ( ∂ u 1 ∂ z u 2 + u 1 ∂ u 2 ∂ z ) k \bold{grad}{u}=\frac{\partial{u}}{\partial{x}}i+\frac{\partial{u}}{\partial{y}}j+\frac{\partial{u}}{\partial{x}}k \\ \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 gradu=∂x∂ui+∂y∂uj+∂x∂uk∂x∂u1u2i=(∂x∂u1u2+u1∂x∂u2)i∂y∂u1u2j=(∂y∂u1u2+u1∂y∂u2)j∂z∂u1u2k=(∂z∂u1u2+u1∂z∂u2)k
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上述3个式子两侧分别相加:
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g r a d u 1 u 2 = u 2 ( ∂ u 1 ∂ x i + ∂ u 1 ∂ y j + ∂ u 1 ∂ z k ) + u 1 ( ∂ u 2 ∂ x i + ∂ u 2 ∂ y j + ∂ u 2 ∂ z k ) = u 1 g r a d u 2 + u 2 g r a d u 1 \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 gradu1u2=u2(∂x∂u1i+∂y∂u1j+∂z∂u1k)+u1(∂x∂u2i+∂y∂u2j+∂z∂u2k)=u1gradu2+u2gradu1
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g r a d F ( u ) = F ′ ( u ) g r a d u \bold{grad}F(u)=F'(u)\bold{grad}{u} gradF(u)=F′(u)gradu
- g r a d F ( u ) = ∂ F ( u ) ∂ x i + ∂ F ( u ) ∂ y j + ∂ F ( u ) ∂ x k = ∂ F ( u ) ∂ u ∂ u ∂ x i + ∂ F ( u ) ∂ u ∂ u ∂ y j + ∂ F ( u ) ∂ u ∂ u ∂ z k = ∂ F ( u ) ∂ u ( ∂ u ∂ x i + ∂ u ∂ y j + ∂ u ∂ x k ) = ∂ F ( u ) ∂ u g r a d u = F ′ ( u ) g r a d u \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 =F'(u)\bold{grad}u gradF(u)=∂x∂F(u)i+∂y∂F(u)j+∂x∂F(u)k=∂u∂F(u)∂x∂ui+∂u∂F(u)∂y∂uj+∂u∂F(u)∂z∂uk=∂u∂F(u)(∂x∂ui+∂y∂uj+∂x∂uk)=∂u∂F(u)gradu=F′(u)gradu