# 16.12: Partial Derivatives

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• $$\frac{\partial^2 z}{\partial x \partial y}=\frac{\partial^2 z}{\partial y \partial x}$$
• $$\left(\frac{\partial y}{\partial x}\right)_{z,u}=\frac{1}{\left(\partial x/\partial y\right)_{z,u}}$$
• $$\left(\frac{\partial y}{\partial x}\right)_{z}\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z}{\partial y}\right)_{x}=-1$$
• $$du=\left(\frac{\partial u}{\partial x_1}\right)_{x_2,x_3...}dx_1+\left(\frac{\partial u}{\partial x_2}\right)_{x_1,x_3...}dx_2+\left(\frac{\partial u}{\partial x_3}\right)_{x_1,x_2...}dx_3$$
• Given $$u=u(x,y)$$, $$x=x(\theta,r)$$ and $$y=y(\theta,r)$$

$\begin{array}{c} \left ( \frac{\partial u}{\partial r} \right )_\theta=\left ( \frac{\partial u}{\partial x} \right )_y\left ( \frac{\partial x}{\partial r} \right )_\theta+\left ( \frac{\partial u}{\partial y} \right )_x\left ( \frac{\partial y}{\partial r} \right )_\theta \\ \left ( \frac{\partial u}{\partial \theta} \right )_r=\left ( \frac{\partial u}{\partial x} \right )_y\left ( \frac{\partial x}{\partial \theta} \right )_r+\left ( \frac{\partial u}{\partial y} \right )_x\left ( \frac{\partial y}{\partial \theta} \right )_r \end{array} \nonumber$

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