The question goes like this:
Use the chain rule to verify the identity y * (∂g/∂x) + x * (∂g/∂y) = 0 if g(x,y) = f(x^2 - y^2, y^2 - x^2) and f is differentiable.
No answer key to help, so any support would be appreciated, thanks!
The question goes like this:
Use the chain rule to verify the identity y * (∂g/∂x) + x * (∂g/∂y) = 0 if g(x,y) = f(x^2 - y^2, y^2 - x^2) and f is differentiable.
No answer key to help, so any support would be appreciated, thanks!
Denote $\displaystyle u=x^2-y^2,\;v=y^2-x^2.$ Then,
$\displaystyle \begin{Bmatrix} \dfrac{{\partial g}}{{\partial x}}=\dfrac{{\partial f}}{{\partial u}}\dfrac{{\partial u}}{{\partial x}}+\dfrac{{\partial f}}{{\partial v}}\dfrac{{\partial v}}{{\partial x}}=\dfrac{{\partial f}}{{\partial u}}(2x)+\dfrac{{\partial f}}{{\partial v}}(-2x)\\ \\\dfrac{{\partial g}}{{\partial y}}=\dfrac{{\partial f}}{{\partial u}}\dfrac{{\partial u}}{{\partial y}}+\dfrac{{\partial f}}{{\partial v}}\dfrac{{\partial v}}{{\partial y}}=\dfrac{{\partial f}}{{\partial u}}(-2y)+\dfrac{{\partial f}}{{\partial v}}(2y) \end{matrix}$