Partial second order differential proof

[ProPatto16]

Homework Statement

if u=f(x,y) where x=e^s*cost and y=e^s*sint show that

(d²u/dx²)+(d²u/dy²) = e^-2s[(d²u/ds²)+(d²u/dt²)

Homework Equations

chain rule relevant here for du/ds and du/dt is

du/ds=du/dx*dx/s+du/dy*dy/ds and replace s with t for du/dt

The Attempt at a Solution

for starters, I am fairly certain this proof can be shown by evaluating both sides and getting them equal. however, just a couple of questions about how to evaluate.

- for du/dx, i take derivitive of u with respect to x, so does that mean i take the derivitive of the function e^scost deriving variables s and t?

- for the double derivitives on the RHS that i need the chain rule for. to take the second partial derivitive do i just take du/ds of the solution to du/ds?

hope you get what i mean :/

 

[mighty2000]

Hello! Yes, you are correct that the proof can be shown by evaluating both sides and getting them equal. To answer your questions:

- Yes, to find du/dx, you would take the derivative of u with respect to x. This means you would take the derivative of the function u(s,t) with respect to s, while treating t as a constant. So you would use the chain rule to take the derivative of e^s*cost with respect to s, while treating t as a constant.
- For the double derivatives on the RHS, you are correct that you would need to use the chain rule. To take the second partial derivative, you would first find the first partial derivative (using the chain rule if needed) and then take the derivative of that result with respect to the variable you are interested in. So for (d^2u/ds^2), you would first find (du/ds) and then take the derivative of that result with respect to s.