多元微积分概念总结。
复合函数的中间变量均为一元函数
设
\[z = f(u, v), \quad \begin{cases} u = g(t) \\ v = h(t) \end{cases}\]则
\[\dfrac{dz}{dt} = \left[\dfrac{\partial z}{\partial u} \; \dfrac{\partial z}{\partial v}\right] \begin{bmatrix} \dfrac{du}{dt} \\ \dfrac{dv}{dt} \end{bmatrix} = \dfrac{\partial z}{\partial u}\dfrac{du}{dt} + \dfrac{\partial z}{\partial v}\dfrac{dv}{dt}\]复合函数的中间变量均为多元函数
设
\[z = f(u, v), \quad \begin{cases} u = g(x, y) \\ v = h(x, y) \end{cases}\]则
\[\begin{bmatrix} \dfrac{\partial z}{\partial x} \\ \dfrac{\partial z}{\partial y} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial v}{\partial x} \\ \dfrac{\partial u}{\partial y} & \dfrac{\partial v}{\partial y} \end{bmatrix} \begin{bmatrix} \dfrac{\partial z}{\partial u} \\ \dfrac{\partial z}{\partial v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x} + \dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial x} \\ \dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y} + \dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial y} \end{bmatrix}\]复合函数的中间变量既有一元函数，又有多元函数
设
\[z = f(u, v), \quad \begin{cases} u = g(x, y) \\ v = h(y) \end{cases}\]则
\[\begin{bmatrix} \dfrac{\partial z}{\partial x} \\ \dfrac{\partial z}{\partial y} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial u}{\partial x} & 0 \\ \dfrac{\partial u}{\partial y} & \dfrac{dv}{dy} \end{bmatrix} \begin{bmatrix} \dfrac{\partial z}{\partial u} \\ \dfrac{\partial z}{\partial v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x} \\ \dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y} + \dfrac{\partial z}{\partial v}\dfrac{dv}{dy} \end{bmatrix}\]