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Math Help - x(u,v),y(u,v) to u(x,y),v(x,y)

  1. #1
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    x(u,v),y(u,v) to u(x,y),v(x,y)

    Can anybody help with this:
    I have x=u+sin v, y=v+cos u. How to extract u,v as functions from x,y, i.e. to get u(x,y) and v(x,y)?
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  2. #2
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    Essentially, you want to solve the two non-linear equations x= u+ sin v, y= v+ cos u for u and v in terms of x and y. I doubt that there will be any elementary solution for that.
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  3. #3
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    yes, i believe in that, so i am looking for some particular step which could make this task easier.
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  4. #4
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    Does a problem specifically ask for u(x,y) and v(x,y)?

    Because if it is only asking for derivatives at certain points, you don't actually need the explicit formulas.
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  5. #5
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    No, the task actually is:
    x=u+sin(v)
    y=v+cos(u)
    z(u,v)=uv^2
    Find \frac{\partial z}{\partial x} + \frac {\partial z}{\partial y} at point (\frac {\pi +3}{3} , \frac {\pi +1}{3}).
    So I must find \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}, \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}, \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}, \frac{\partial z}{\partial v} \frac{\partial v}{\partial y}.
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  6. #6
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    Right, so you don't actually need the explicit formula.

    For example, to find \frac{\partial u}{\partial x} and \frac{\partial v}{\partial x}.

    Take the partials of the equations directly (independent variables are x,y):
    <br />
\frac{\partial x}{\partial x} = \frac{\partial u}{\partial x} + \cos(v)\frac{\partial v}{\partial x}<br />

    <br />
\frac{\partial y}{\partial x} = \frac{\partial v}{\partial x} - \sin(u)\frac{\partial u}{\partial x}<br />

    Note that \frac{\partial x}{\partial x} = 1 and \frac{\partial y}{\partial x}=0.

    Solve this system of linear equations at the given point for \frac{\partial v}{\partial x}, \frac{\partial u}{\partial x}.

    Similar process for other derivatives.

    Note you do need to solve for u and v at the point (\frac {\pi +3}{3} , \frac {\pi +1}{3}), and you use these values when evaluating things like sin(u).
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  7. #7
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    Quote Originally Posted by waytogo View Post
    No, the task actually is:
    x=u+sin(v)
    y=v+cos(u)
    z(u,v)=uv^2
    Find \frac{\partial z}{\partial x} + \frac {\partial z}{\partial y} at point (\frac {\pi +3}{3} , \frac {\pi +1}{3}).
    So I must find \frac{\partial z}{\partial u} \frac{\partial u}{\partial x}, \frac{\partial z}{\partial u} \frac{\partial u}{\partial y}, \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}, \frac{\partial z}{\partial v} \frac{\partial v}{\partial y}.
    Do you know about Jacobians? I would suggest you check the point. It looks a little dubious.
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  8. #8
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    Is \left(\frac{\pi+ 3}{3}, \frac{\pi+ 1}{3}\right) (x, y) or (u, v)?
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  9. #9
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    the point was incorrect, the actual point is (\frac{\pi+3}{3},\frac{\pi+1}{2}).
    If I have caught your idea, I should use theorem about the inverse Jacobian?
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  10. #10
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    The point now makes sense. Yes, inverse Jacobians is the way I would do this.
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