全微分形式不变性 设函数$z = f\left( {u,v} \right)$具有连续偏导数，则有全微分

$${\text{d}}z = \frac{{\partial z}}{{\partial u}}{\text{d}}u + \frac{{\partial z}}{{\partial v}}{\text{d}}v$$

如果$v,u$又是中间变量，即$u = \varphi \left( {x,y} \right),v = \phi \left( {x,y} \right)$，且这两个函数也具有连续偏导数，则

\[\begin{gathered} {\text{d}}u = \frac{{u\varphi }}{{\partial x}}{\text{d}}x + \frac{{\partial u}}{{\partial y}}{\text{d}}y, \hfill \\ {\text{d}}v = \frac{{\partial v}}{{\partial x}}{\text{d}}x + \frac{{\partial v}}{{\partial y}}{\text{d}}y \hfill \\ \end{gathered} \]

复合函数

$$z = f\left[ {\varphi \left( {x,y} \right),\phi \left( {x,y} \right)} \right]$$

的全微分为

\[\begin{gathered} {\text{d}}z = \frac{{\partial z}}{{\partial u}}{\text{d}}u + \frac{{\partial z}}{{\partial v}}{\text{d}}v \hfill \\ = \frac{{\partial z}}{{\partial u}}\left( {\frac{{\partial u}}{{\partial x}}{\text{d}}x + \frac{{\partial u}}{{\partial y}}{\text{d}}y} \right) + \hfill \\ \frac{{\partial z}}{{\partial v}}\left( {\frac{{\partial v}}{{\partial x}}{\text{d}}x + \frac{{\partial v}}{{\partial y}}{\text{d}}y} \right) \hfill \\ = \frac{{\partial z}}{{\partial u}}\frac{{\partial u}}{{\partial x}}{\text{d}}x + \frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial x}}{\text{d}}x + \hfill \\ \frac{{\partial z}}{{\partial u}}\frac{{\partial u}}{{\partial y}}{\text{d}}y + \frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial y}}{\text{d}}y \hfill \\ = \left( {\frac{{\partial z}}{{\partial u}}\frac{{\partial u}}{{\partial x}} + \frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial x}}} \right){\text{d}}x + \hfill \\ \left( {\frac{{\partial z}}{{\partial u}}\frac{{\partial u}}{{\partial y}} + \frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial y}}} \right){\text{d}}y \hfill \\ = \frac{{\partial z}}{{\partial x}}{\text{d}}x + \frac{{\partial z}}{{\partial y}}{\text{d}}y \hfill \\ \end{gathered} \]

由此可见，无论$v,u$是自变量还是中间变量，函数$z = f\left( {u,v} \right)$的全微分形式是一样的.这个性质叫做全微分形式不变性.