【高等数学】多元微分学(二)
隐函数的偏导数
二元方程的隐函数
F ( x , y ) = 0 F(x,y)=0 F(x,y)=0
推出隐函数形式 y = y ( x ) y=y(x) y=y(x). 欲求 d y d x \frac{d y}{d x} dxdy 需要对 F = 0 F=0 F=0 两边同时对 x x x 求全导
0 = d d x F ( x , y ( x ) ) = ∂ F ∂ x d x d x + ∂ F ∂ y d y d x = ∂ F ∂ x d x d x + ∂ F ∂ y d y d x 0 = \frac{d}{dx} F(x,y(x))= \frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx}=\frac{\partial F}{\partial x} \frac{dx}{dx}+\frac{\partial F}{\partial y} \frac{dy}{dx} 0=dxdF(x,y(x))=∂x∂Fdxdx+∂y∂Fdxdy=∂x∂Fdxdx+∂y∂Fdxdy
求出 d y d x = − ∂ F ∂ x ∂ F ∂ y \frac{dy}{dx}= -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}} dxdy=−∂y∂F∂x∂F
三元方程(组)的隐函数
三元方程
F ( x , y , z ) = 0 F(x,y,z)=0 F(x,y,z)=0,
推出隐函数形式 z = z ( x , y ) z=z(x,y) z=z(x,y). 欲求 ∂ z ∂ x \frac{\partial z}{\partial x} ∂x∂z 需要对 F = 0 F=0 F=0 两边同时对 x x x 求偏导
0 = ∂ F ∂ x + ∂ F ∂ z ∂ z ∂ x 0=\frac{\partial F}{\partial x} +\frac{\partial F}{\partial z} \frac{\partial z}{\partial x} 0=∂x∂F+∂z∂F∂x∂z
求出 ∂ z ∂ x = − ∂ F ∂ x ∂ F ∂ z \frac{\partial z}{\partial x}= -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}} ∂x∂z=−∂z∂F∂x∂F
两个三元方程
F ( x , y , z ) = 0 F(x,y,z)=0 F(x,y,z)=0, G ( x , y , z ) = 0 G(x,y,z)=0 G(x,y,z)=0
推出隐函数形式 z = z ( x ) , y = y ( x ) z=z(x), y=y(x) z=z(x),y=y(x) 欲求 d z d x \frac{d z}{dx} dxdz 需要对 F = 0 , G = 0 F=0,G=0 F=0,G=0 两边同时对 x x x 求全导
0 = ∂ F ∂ x + ∂ F ∂ y d y d x + ∂ F ∂ z d z d x 0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}\frac{dy}{dx}+\frac{\partial F}{\partial z}\frac{dz}{dx} 0=∂x∂F+∂y∂Fdxdy+∂z∂Fdxdz
0 = ∂ G ∂ x + ∂ G ∂ y d y d x + ∂ G ∂ z d z d x 0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial y}\frac{dy}{dx}+\frac{\partial G}{\partial z}\frac{dz}{dx} 0=∂x∂G+∂y∂Gdxdy+∂z∂Gdxdz
根据克拉姆法则,
J = ∂ ( F , G ) ∂ ( y , z ) = ∣ ∂ F ∂ y ∂ F ∂ z ∂ G ∂ y ∂ G ∂ z ∣ J=\frac{\partial (F,G)}{\partial(y,z)}= \left|\begin{array}{c c} \frac{\partial F}{\partial y} &\frac{\partial F}{\partial z} \\\\ \frac{\partial G}{\partial y} &\frac{\partial G}{\partial z}\end{array}\right| J=∂(y,z)∂(F,G)= ∂y∂F∂y∂G∂z∂F∂z∂G
d y d x = − 1 J ∂ ( F , G ) ∂ ( x , z ) \frac{d y}{d x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (x,z)} dxdy=−J1∂(x,z)∂(F,G) d z d x = − 1 J ∂ ( F , G ) ∂ ( y , x ) \frac{d z}{d x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (y,x)} dxdz=−J1∂(y,x)∂(F,G)
四元方程(组)的隐函数
四元函数方程
F ( x , y , z , w ) = 0 F(x,y,z,w)=0 F(x,y,z,w)=0,
推出隐函数形式 w = w ( x , y , z ) w=w(x,y,z) w=w(x,y,z)
0 = ∂ F ∂ x + ∂ F ∂ w ∂ w ∂ x 0= \frac{\partial F}{\partial x}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial x} 0=∂x∂F+∂w∂F∂x∂w 求出
∂ w ∂ x = − ∂ F ∂ x ∂ F ∂ w \frac{\partial w}{\partial x}= - \frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial w}} ∂x∂w=−∂w∂F∂x∂F 类似的
∂ w ∂ y = − ∂ F ∂ y ∂ F ∂ w \frac{\partial w}{\partial y}= - \frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial w}} ∂y∂w=−∂w∂F∂y∂F ∂ w ∂ z = − ∂ F ∂ z ∂ F ∂ w \frac{\partial w}{\partial z}= - \frac{\frac{\partial F}{\partial z}}{\frac{\partial F}{\partial w}} ∂z∂w=−∂w∂F∂z∂F
两个四元函数方程组
F ( x , y , z , w ) = 0 F(x,y,z,w)=0 F(x,y,z,w)=0, G ( x , y , z , w ) = 0 G(x,y,z,w)=0 G(x,y,z,w)=0 推出隐函数形式 z = z ( x , y ) z=z(x,y) z=z(x,y), w = w ( x , y ) w=w(x,y) w=w(x,y)
0 = ∂ F ∂ x + ∂ F ∂ z ∂ z ∂ x + ∂ F ∂ w ∂ w ∂ x 0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial z} \frac{\partial z}{\partial x}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial x} 0=∂x∂F+∂z∂F∂x∂z+∂w∂F∂x∂w
0 = ∂ G ∂ x + ∂ G ∂ z ∂ z ∂ x + ∂ G ∂ w ∂ w ∂ x 0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial z} \frac{\partial z}{\partial x}+\frac{\partial G}{\partial w} \frac{\partial w}{\partial x} 0=∂x∂G+∂z∂G∂x∂z+∂w∂G∂x∂w
根据克拉姆法则, J = ∂ ( F , G ) ∂ ( z , w ) = ∣ ∂ F ∂ z ∂ F ∂ w ∂ G ∂ z ∂ G ∂ w ∣ J=\frac{\partial (F,G)}{\partial(z,w)}= \left|\begin{matrix}\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w}\\\\\frac{\partial G}{\partial z} &\frac{\partial G}{\partial w}\end{matrix}\right| J=∂(z,w)∂(F,G)= ∂z∂F∂z∂G∂w∂F∂w∂G
∂ z ∂ x = − 1 J ∂ ( F , G ) ∂ ( x , w ) \frac{\partial z}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (x,w)} ∂x∂z=−J1∂(x,w)∂(F,G) ∂ w ∂ x = − 1 J ∂ ( F , G ) ∂ ( z , x ) \frac{\partial w}{\partial x} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (z,x)} ∂x∂w=−J1∂(z,x)∂(F,G)
类似的由 0 = ∂ F ∂ y + ∂ F ∂ z ∂ z ∂ y + ∂ F ∂ w ∂ w ∂ y 0=\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z} \frac{\partial z}{\partial y}+\frac{\partial F}{\partial w} \frac{\partial w}{\partial y} 0=∂y∂F+∂z∂F∂y∂z+∂w∂F∂y∂w 0 = ∂ G ∂ y + ∂ G ∂ z ∂ z ∂ y + ∂ G ∂ w ∂ w ∂ y 0=\frac{\partial G}{\partial y}+\frac{\partial G}{\partial z} \frac{\partial z}{\partial y}+\frac{\partial G}{\partial w} \frac{\partial w}{\partial y} 0=∂y∂G+∂z∂G∂y∂z+∂w∂G∂y∂w 得到
∂ z ∂ y = − 1 J ∂ ( F , G ) ∂ ( y , w ) \frac{\partial z}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (y,w)} ∂y∂z=−J1∂(y,w)∂(F,G) ∂ w ∂ y = − 1 J ∂ ( F , G ) ∂ ( z , y ) \frac{\partial w}{\partial y} = -\frac{1}{J} \frac{\partial (F, G)}{\partial (z,y)} ∂y∂w=−J1∂(z,y)∂(F,G)
三个四元函数方程组
F ( x , y , z , w ) = 0 F(x,y,z,w)=0 F(x,y,z,w)=0, G ( x , y , z , w ) = 0 G(x,y,z,w)=0 G(x,y,z,w)=0, H ( x , y , z , w ) = 0 H(x,y,z,w)=0 H(x,y,z,w)=0.
推出隐函数形式 y = y ( x ) , z = z ( x ) , w = w ( x ) y=y(x), z=z(x), w=w(x) y=y(x),z=z(x),w=w(x)
0 = ∂ F ∂ x + ∂ F ∂ y d y d x + ∂ F ∂ z d z d x + ∂ F ∂ w d w d x 0=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} \frac{d y}{d x}+\frac{\partial F}{\partial z} \frac{d z}{d x}+\frac{\partial F}{\partial w} \frac{d w}{d x} 0=∂x∂F+∂y∂Fdxdy+∂z∂Fdxdz+∂w∂Fdxdw
0 = ∂ G ∂ x + ∂ G ∂ y d y d x + ∂ G ∂ z d z d x + ∂ G ∂ w d w d x 0=\frac{\partial G}{\partial x}+\frac{\partial G}{\partial y} \frac{d y}{d x}+\frac{\partial G}{\partial z} \frac{d z}{d x}+\frac{\partial G}{\partial w} \frac{d w}{d x} 0=∂x∂G+∂y∂Gdxdy+∂z∂Gdxdz+∂w∂Gdxdw
0 = ∂ H ∂ x + ∂ H ∂ y d y d x + ∂ H ∂ z d z d x + ∂ H ∂ w d w d x 0=\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y} \frac{d y}{d x}+\frac{\partial H}{\partial z} \frac{d z}{d x}+\frac{\partial H}{\partial w} \frac{d w}{d x} 0=∂x∂H+∂y∂Hdxdy+∂z∂Hdxdz+∂w∂Hdxdw
由克拉姆法则,
J = ∂ ( F , G , H ) ∂ ( y , z , w ) = ∣ ∂ F ∂ y ∂ F ∂ z ∂ F ∂ w ∂ G ∂ y ∂ G ∂ z ∂ G ∂ w ∂ H ∂ y ∂ H ∂ z ∂ H ∂ w ∣ J= \frac{\partial(F, G, H)}{\partial (y,z,w)} = \left| \begin{array}{c c c} \frac{\partial F}{\partial y} &\frac{\partial F}{\partial z} & \frac{\partial F}{\partial w} \\\\ \frac{\partial G}{\partial y}&\frac{\partial G}{\partial z} &\frac{\partial G}{\partial w} \\\\ \frac{\partial H}{\partial y}&\frac{\partial H}{\partial z} &\frac{\partial H}{\partial w}\end{array}\right| J=∂(y,z,w)∂(F,G,H)= ∂y∂F∂y∂G∂y∂H∂z∂F∂z∂G∂z∂H∂w∂F∂w∂G∂w∂H
d y d x = − 1 J ∂ ( F , G , H ) ∂ ( x , z , w ) \frac{d y}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (x,z,w)} dxdy=−J1∂(x,z,w)∂(F,G,H) d z d x = − 1 J ∂ ( F , G , H ) ∂ ( y , x , w ) \frac{d z}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (y,x,w)} dxdz=−J1∂(y,x,w)∂(F,G,H) d w d x = − 1 J ∂ ( F , G , H ) ∂ ( y , z , x ) \frac{d w}{d x}=-\frac{1}{J} \frac{\partial (F,G,H)}{\partial (y,z,x)} dxdw=−J1∂(y,z,x)∂(F,G,H)
五元方程
两个五元方程
F ( x , y , z , u , v ) = 0 F(x,y,z,u,v)=0 F(x,y,z,u,v)=0, G ( x , y , z , u , v ) = 0 G(x,y,z,u,v)=0 G(x,y,z,u,v)=0
推出隐函数形式 u = u ( x , y , z ) , v = v ( x , y , z ) u=u(x,y,z), v=v(x,y,z) u=u(x,y,z),v=v(x,y,z), 对 F = 0 , G = 0 F=0, G=0 F=0,G=0 同时对 x x x 求偏微分
0 = ∂ F ∂ x + ∂ F ∂ u ∂ u ∂ x + ∂ F ∂ v ∂ v ∂ x 0= \frac{\partial F}{\partial x} + \frac{\partial F}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial F}{\partial v}\frac{\partial v}{\partial x} 0=∂x∂F+∂u∂F∂x∂u+∂v∂F∂x∂v 0 = ∂ G ∂ x + ∂ G ∂ u ∂ u ∂ x + ∂ G ∂ v ∂ v ∂ x 0= \frac{\partial G}{\partial x} + \frac{\partial G}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial G}{\partial v}\frac{\partial v}{\partial x} 0=∂x∂G+∂u∂G∂x∂u+∂v∂G∂x∂v
由克拉姆法则 J = ∂ ( F , G ) ∂ ( u , v ) J=\frac{\partial(F,G)}{\partial (u,v)} J=∂(u,v)∂(F,G), ∂ u ∂ x = − 1 J ∂ ( F , G ) ∂ ( x , v ) \frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(x,v)} ∂x∂u=−J1∂(x,v)∂(F,G) ∂ v ∂ x = − 1 J ∂ ( F , G ) ∂ ( u , x ) \frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,x)} ∂x∂v=−J1∂(u,x)∂(F,G)
类似的
∂ u ∂ y = − 1 J ∂ ( F , G ) ∂ ( y , v ) \frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(y,v)} ∂y∂u=−J1∂(y,v)∂(F,G) ∂ u ∂ y = − 1 J ∂ ( F , G ) ∂ ( u , y ) \frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,y)} ∂y∂u=−J1∂(u,y)∂(F,G)
∂ u ∂ z = − 1 J ∂ ( F , G ) ∂ ( z , v ) \frac{\partial u}{\partial z}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(z,v)} ∂z∂u=−J1∂(z,v)∂(F,G) ∂ u ∂ z = − 1 J ∂ ( F , G ) ∂ ( u , z ) \frac{\partial u}{\partial z}=-\frac{1}{J}\frac{\partial(F,G)}{\partial(u,z)} ∂z∂u=−J1∂(u,z)∂(F,G)
三个五元方程
F ( x , y , z , u , v ) = 0 F(x,y,z,u,v)=0 F(x,y,z,u,v)=0, G ( x , y , z , u , v ) = 0 G(x,y,z,u,v)=0 G(x,y,z,u,v)=0, H ( x , y , z , u , v ) = 0 H(x,y,z,u,v)=0 H(x,y,z,u,v)=0
推出隐函数形式 z = z ( x , y ) , u = u ( x , y ) , v = v ( x , y ) z=z(x,y), u=u(x,y), v=v(x,y) z=z(x,y),u=u(x,y),v=v(x,y)
0 = ∂ F ∂ x + ∂ F ∂ z ∂ z ∂ x + ∂ F ∂ u ∂ u ∂ x + ∂ F ∂ v ∂ v ∂ x 0= \frac{\partial F}{\partial x} +\frac{\partial F}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial F}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial F}{\partial v}\frac{\partial v}{\partial x} 0=∂x∂F+∂z∂F∂x∂z+∂u∂F∂x∂u+∂v∂F∂x∂v
0 = ∂ G ∂ x + ∂ G ∂ z ∂ z ∂ x + ∂ G ∂ u ∂ u ∂ x + ∂ G ∂ v ∂ v ∂ x 0= \frac{\partial G}{\partial x} +\frac{\partial G}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial G}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial G}{\partial v}\frac{\partial v}{\partial x} 0=∂x∂G+∂z∂G∂x∂z+∂u∂G∂x∂u+∂v∂G∂x∂v
0 = ∂ H ∂ x + ∂ H ∂ z ∂ z ∂ x + ∂ H ∂ u ∂ u ∂ x + ∂ H ∂ v ∂ v ∂ x 0= \frac{\partial H}{\partial x} +\frac{\partial H}{\partial z}\frac{\partial z}{\partial x}+ \frac{\partial H}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial H}{\partial v}\frac{\partial v}{\partial x} 0=∂x∂H+∂z∂H∂x∂z+∂u∂H∂x∂u+∂v∂H∂x∂v
由克拉姆法则 J = ∂ ( F , G , H ) ∂ ( z , u , v ) J=\frac{\partial(F,G,H)}{\partial (z,u,v)} J=∂(z,u,v)∂(F,G,H),
∂ z ∂ x = − 1 J ∂ ( F , G , H ) ∂ ( x , u , v ) \frac{\partial z}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(x,u,v)} ∂x∂z=−J1∂(x,u,v)∂(F,G,H) ∂ u ∂ x = − 1 J ∂ ( F , G , H ) ∂ ( z , x , v ) \frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,x,v)} ∂x∂u=−J1∂(z,x,v)∂(F,G,H) ∂ v ∂ x = − 1 J ∂ ( F , G , H ) ∂ ( z , u , x ) \frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,u,x)} ∂x∂v=−J1∂(z,u,x)∂(F,G,H)
类似的 ∂ z ∂ y = − 1 J ∂ ( F , G , H ) ∂ ( y , u , v ) \frac{\partial z}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(y,u,v)} ∂y∂z=−J1∂(y,u,v)∂(F,G,H) ∂ u ∂ y = − 1 J ∂ ( F , G , H ) ∂ ( z , y , v ) \frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,y,v)} ∂y∂u=−J1∂(z,y,v)∂(F,G,H) ∂ v ∂ y = − 1 J ∂ ( F , G , H ) ∂ ( z , u , y ) \frac{\partial v}{\partial y}=-\frac{1}{J}\frac{\partial(F,G,H)}{\partial(z,u,y)} ∂y∂v=−J1∂(z,u,y)∂(F,G,H)
(选看) 反函数的偏导数
一元一维函数 y = f ( x ) y=f(x) y=f(x)
反函数为
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x
∂
u
\frac{\partial v}{\partial y}=\frac{1}{J}\frac{\partial x}{\partial u}
∂y∂v=J1∂u∂x
用线性代数矩阵的乘法表示
[ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ] [ ∂ u ∂ x ∂ u ∂ y ∂ v ∂ x ∂ v ∂ y ] = [ 1 0 0 1 ] \left[\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix}\right]\left[\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{matrix}\right] =\left[\begin{matrix} 1 & 0\\ 0& 1 \end{matrix}\right] [∂u∂x∂u∂y∂v∂x∂v∂y][∂x∂u∂x∂v∂y∂u∂y∂v]=[1001]
三元三维函数 u = ( x , y , z ) , v ( x , y , z ) , w ( x , y , z ) u=(x,y,z), v(x,y,z), w(x,y,z) u=(x,y,z),v(x,y,z),w(x,y,z)
类似地
[ ∂ x ∂ u ∂ x ∂ v ∂ x ∂ w ∂ y ∂ u ∂ y ∂ v ∂ y ∂ w ∂ z ∂ u ∂ z ∂ v ∂ z ∂ w ] [ ∂ u ∂ x ∂ u ∂ y ∂ u ∂ z ∂ v ∂ x ∂ v ∂ y ∂ v ∂ z ∂ w ∂ x ∂ w ∂ y ∂ w ∂ z ] = [ 1 0 0 0 1 0 0 0 1 ] \left[\begin{matrix}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\\ \end{matrix}\right]\left[\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} &\frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} &\frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} &\frac{\partial w}{\partial z}\\ \end{matrix}\right] =\left[\begin{matrix} 1 & 0& 0\\ 0& 1 &0\\ 0& 0 &1 \end{matrix}\right] ∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z ∂x∂u∂x∂v∂x∂w∂y∂u∂y∂v∂y∂w∂z∂u∂z∂v∂z∂w = 100010001
(选看) 参数方程的微分
二维单参数方程 (一元二维)
x
=
x
(
t
)
x=x(t)
x=x(t),
y
=
y
(
t
)
y=y(t)
y=y(t),
d
y
d
x
=
d
y
d
t
d
x
d
t
\frac{d y}{d x}= \frac{\frac{d y}{d t}}{\frac{dx}{dt}}
dxdy=dtdxdtdy
三维单参数方程 (一元三维)
x
=
x
(
t
)
x=x(t)
x=x(t),
y
=
y
(
t
)
y=y(t)
y=y(t),
z
=
z
(
t
)
z=z(t)
z=z(t)
d
y
d
x
=
d
y
d
t
d
x
d
t
\frac{d y}{d x}= \frac{\frac{d y}{d t}}{\frac{dx}{dt}}
dxdy=dtdxdtdy
d
z
d
x
=
d
z
d
t
d
x
d
t
\frac{d z}{d x}= \frac{\frac{d z}{d t}}{\frac{dx}{dt}}
dxdz=dtdxdtdz
d
z
d
y
=
d
z
d
t
d
y
d
t
\frac{d z}{d y}= \frac{\frac{d z}{d t}}{\frac{dy}{dt}}
dydz=dtdydtdz
三维双参数方程 (二元三维)
x = x ( s , t ) x=x(s,t) x=x(s,t), y = y ( s , t ) y=y(s,t) y=y(s,t), z = z ( s , t ) z=z(s,t) z=z(s,t), 求解 ∂ z ∂ x \frac{\partial z}{\partial x} ∂x∂z, ∂ z ∂ y \frac{\partial z}{\partial y} ∂y∂z
推出隐函数 s = s ( x , y ) s=s(x,y) s=s(x,y), t = t ( x , y ) t=t(x,y) t=t(x,y)
z = z ( s , t ) = z ( s ( x , y ) , t ( x , y ) ) z=z(s,t)=z(s(x,y), t(x,y)) z=z(s,t)=z(s(x,y),t(x,y)),
∂ z ∂ x = ∂ z ∂ s ∂ s ∂ x + ∂ z ∂ t ∂ t ∂ x \frac{\partial z}{\partial x}= \frac{\partial z}{\partial s}\frac{\partial s}{\partial x}+ \frac{\partial z}{\partial t}\frac{\partial t}{\partial x} ∂x∂z=∂s∂z∂x∂s+∂t∂z∂x∂t
∂ z ∂ y = ∂ z ∂ s ∂ s ∂ y + ∂ z ∂ t ∂ t ∂ y \frac{\partial z}{\partial y}= \frac{\partial z}{\partial s}\frac{\partial s}{\partial y}+\frac{\partial z}{\partial t}\frac{\partial t}{\partial y} ∂y∂z=∂s∂z∂y∂s+∂t∂z∂y∂t
结合二元二维反函数
记 J = ∂ ( x , y ) ∂ ( s , t ) J=\frac{\partial (x,y)}{\partial (s,t)} J=∂(s,t)∂(x,y)
∂
s
∂
x
=
1
J
∂
y
∂
t
\frac{\partial s}{\partial x}=\frac{1}{J}\frac{\partial y}{\partial t}
∂x∂s=J1∂t∂y
∂
t
∂
x
=
−
1
J
∂
y
∂
s
\frac{\partial t}{\partial x}=-\frac{1}{J}\frac{\partial y}{\partial s}
∂x∂t=−J1∂s∂y
∂
s
∂
y
=
−
1
J
∂
x
∂
t
\frac{\partial s}{\partial y}=-\frac{1}{J}\frac{\partial x}{\partial t}
∂y∂s=−J1∂t∂x
∂
t
∂
y
=
1
J
∂
x
∂
s
\frac{\partial t}{\partial y}=\frac{1}{J}\frac{\partial x}{\partial s}
∂y∂t=J1∂s∂x
∂
z
∂
x
=
1
J
∂
(
z
,
x
)
∂
(
s
,
t
)
\frac{\partial z}{\partial x} = \frac{1}{J} \frac{\partial (z,x)}{\partial (s,t)}
∂x∂z=J1∂(s,t)∂(z,x)
∂
z
∂
y
=
1
J
∂
(
x
,
z
)
∂
(
s
,
t
)
\frac{\partial z}{\partial y} = \frac{1}{J} \frac{\partial (x,z)}{\partial (s,t)}
∂y∂z=J1∂(s,t)∂(x,z)
原文地址:https://blog.csdn.net/serpenttom/article/details/142992026
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