Advanced Calculus -- Finding the local inverses of f

Problem: For each of the following transformations (u,v) = f(x, y): (i) compute the Jacobian det Df, (ii) draw a sketch of the images of some of the lines x = constant and y = constant in the uv-plane, and (iii) find the formulas for the local inverses of f when they exist.

(a) u = (e^x)(cosy), v = (e^x)(siny)

----------------------------------------------------------------------------

I've completed (i) and (ii) but I'm stuck on (iii). I know I have to solve for x and y but don't know how to start. Hints?

Re: Advanced Calculus -- Finding the local inverses of f

I'm tempted to divide by e^x (which never is zero), and then square.

Re: Advanced Calculus -- Finding the local inverses of f

Quote:

Originally Posted by

**TKHunny** I'm tempted to divide by e^x (which never is zero), and then square.

Hmmmm....well, I'm supposed to get x = x(u,v) and y = y(u,v) so if do as such, I get:

u = (e^x)(cosy)

u/(e^x) = cosy

I take the arccos and get

If I solve v for x and try to substitute, it just starts looking troublesome.

Re: Advanced Calculus -- Finding the local inverses of f

Quote:

Originally Posted by

**MissMousey** Problem: For each of the following transformations (u,v) = f(x, y): (i) compute the Jacobian det Df, (ii) draw a sketch of the images of some of the lines x = constant and y = constant in the uv-plane, and (iii) find the formulas for the local inverses of f when they exist.

(a) u = (e^x)(cosy), v = (e^x)(siny)

----------------------------------------------------------------------------

I've completed (i) and (ii) but I'm stuck on (iii). I know I have to solve for x and y but don't know how to start. Hints?

You may find this about log polar coordinates interesting

Log-polar coordinates - Wikipedia, the free encyclopedia

Re: Advanced Calculus -- Finding the local inverses of f

Ok, I got it. Thank you, everyone.