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Math Help - Advanced Calculus -- Finding the local inverses of f

  1. #1
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    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?
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    Re: Advanced Calculus -- Finding the local inverses of f

    I'm tempted to divide by e^x (which never is zero), and then square.
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    Re: Advanced Calculus -- Finding the local inverses of f

    Quote Originally Posted by TKHunny View Post
    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
    arccos (u/e^x) = y

    If I solve v for x and try to substitute, it just starts looking troublesome.
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    Re: Advanced Calculus -- Finding the local inverses of f

    Quote Originally Posted by MissMousey View Post
    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
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  5. #5
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    Re: Advanced Calculus -- Finding the local inverses of f

    Ok, I got it. Thank you, everyone.
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