proof
 

z = f(x+y,x-y), and z has continuous partial derivatives with respect to u = x+y and v = x-y
show that (dz/dx)(dz/dy) = (dz/du)^2 - (dz/dv)^2

chain rule ครับ โป้งเดียวจอด

$\dfrac{\partial z}{\partial x}=\dfrac{\partial f}{\partial u}\cdot \dfrac{\partial u}{\partial x}+\dfrac{\partial f}{\partial v}\cdot \dfrac{\partial v}{\partial x}$

$\dfrac{\partial z}{\partial y}=\dfrac{\partial f}{\partial u}\cdot \dfrac{\partial u}{\partial y}+\dfrac{\partial f}{\partial v}\cdot \dfrac{\partial v}{\partial y}$

อ้ออย่าลืมว่า

$\dfrac{\partial z}{\partial u}=\dfrac{\partial f}{\partial u}$

$\dfrac{\partial z}{\partial v}=\dfrac{\partial f}{\partial v}$

นะครับ