Question

if \[\phi(u,v)=f(x,y)\]where\[u=y^2-x^2,v=y^2+x^2\]

Answer 1

prove that
\[\large \color{blue}{\frac{1}{4xy}\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2\phi}{\partial v^2}-\frac{\partial^2\phi}{\partial u^2}}\]

Answer 2

using \[f_x=\frac{\partial f}{\partial x}\]and the multivariable chain rule we have\[f_x=\phi_uu_x+\phi_vv_x=\phi_u(-2x)+\phi_vv_x(2x)=-2x(\phi_u-\phi_v)\]\[\large f_{xy}=-2x[(\phi_{uu}u_y+\phi_{uv}v_y)-(\phi_{vu}u_y+\phi_{vv}v_y)]\\\large f_{xy}=4xy[(\phi_{vu}+\cancel\phi_{vv})-(\phi_{uu}+\cancel\phi_{uv})]\]from which the result immediately follows

Answer 3

typo :P

Answer 4

I'm curious as to what sort of context this question falls under. This appears to be similar to using a change of basis on a quadratic form like from an ellipse to a circle. But this isn't quite that and it just sort of interests me where this kind of thing would pop up.

Answer 5

\[\large f_{xy}=-2x[(\phi_{uu}u_y+\phi_{uv}v_y)-(\phi_{vu}u_y+\phi_{vv}v_y)]\\\large f_{xy}=4xy[(\cancel\phi_{vu}+\phi_{vv})-(\phi_{uu}+\cancel\phi_{uv})]\]

Answer 6

hm... say you had 4 points of electric potential in an oddly-shaped 4-sided figure and wanted to calculate the electric field at each point between them. or had tracked an object by measuring velocities as it moved along a strange, 4-sided path.

with the right transformation you could simplify the path, and but it seems would need to use this version of the chain rule in order to calculate acceleration, work, etc.

I now that sounds a bit contrived, but I'm just pulling up some random idea :P