Hi, I have a function f(x,y). u = ax + by and v = bx - ay. a and b are constants. i need to prove that d2f/du2 + d2f/dv2 = 0.

I have started out by finding df/du using chain rule:

df/du = df/dx * dx/du + df/dy * dy/du

Hence, df/du = df/a*dx + df/b*dy

My problem is that now i need to find d^2f/du^2 but i do not know how to continue from df/du (ie. i do not know how i can differentiate df/a*dx + df/b*dy with respect to u).

Thanks for reading

 

There is something missing. I can see how u and v are functions of x and y, BUT how is f(x,y) connected to u and v?

So in order to answer the question you need to show three explicit functional relationships and you have only shown two.

Second, be careful with your notation:

df/a*dx + df/b*dy​

is a meaningless construct. You can't mix the differential operator like an ordinary algebraic symbol.

 

i miswrote what i need to prove and have edited it. i actually need to prove that d2f/du2 + d2f/dv2 = 0.

 

You still haven't established that f(x,y) has ANY dependence on u and v. As near as I can tell you haven't done squat to your original post.