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    So x = rcosht, y = rsinht
    d(x,y)/d(r,t) = r, correct?

    But.. for d(r,t)/d(x,y) I can't get 1/r. I end up with r(x^2 - y^2)/x^2.y^2

    Help :/
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    Probably best if you put up a picture of your exact question.
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    dx/dr = cosht
    dx/dt = rsinht

    dy/dr = sinht
    dy/dt = rcosht

    Jacobian = rcosh^2(t) - rsinh^2(t) = r

    What else do you need?
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    (Original post by Zacken)
    Probably best if you put up a picture of your exact question.
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    (Original post by Math12345)
    dx/dr = cosht
    dx/dt = rsinht

    dy/dr = sinht
    dy/dt = rcosht

    Jacobian = rcosh^2(t) - rsinh^2(t) = r

    What else do you need?
    The inverse
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    (Original post by StarvingAutist)
    So x = rcosht, y = rsinht
    d(x,y)/d(r,t) = r, correct?

    But.. for d(r,t)/d(x,y) I can't get 1/r. I end up with r(x^2 - y^2)/x^2.y^2

    Help :/
    Works when I do it.

    For your inverses you can use:

    \displaystyle r=\sqrt{x^2-y^2}\\\\t=\tanh^{-1}\frac{y}{x}

    What do you get for your partial derivatives?
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    (Original post by ghostwalker)
    Works when I do it.

    For your inverses you can use:

    \displaystyle r=\sqrt{x^2-y^2}\\\\t=\tanh^{-1}\frac{y}{x}

    What do you get for your partial derivatives?
    dt/dx = 1/(r sinht)
    dt/dy = 1/(r cosht)
    dr/dx = 1/cosht
    dr/dy = 1/sinht
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    (Original post by StarvingAutist)
    dt/dx = 1/(r sinht)
    dt/dy = 1/(r cosht)
    dr/dx = 1/cosht
    dr/dy = 1/sinht
    I would expect them to be function of x,y, not r,t.

    Looks like you're just taking the reciprocal


    \displaystyle\dfrac{\partial r}{\partial x }=\dfrac{1}{\frac{\partial x}{\partial r}}


    Is that the case?
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    (Original post by ghostwalker)
    I would expect them to be function of x,y, not r,t.

    Looks like you're just taking the reciprocal


    \displaystyle\dfrac{\partial x}{\partial r }=\dfrac{1}{\frac{\partial r}{\partial x}}


    Is that the case?
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    (Original post by StarvingAutist)
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    Starting at the top left, you're taking partial derivatives wrt x with r treated as a constant. But you require partial derivatives wrt x treating y as a constant.

    With fuller notation:

    You're working out \left(\dfrac{\partial \theta}{ \partial x}\right)_r

    But you actually want \left(\dfrac{\partial \theta}{ \partial x}\right)_y

    Hence the need to get r,theta in terms of x,y to start.
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    (Original post by ghostwalker)
    Starting at the top left, you're taking partial derivatives wrt x with r treated as a constant. But you require partial derivatives wrt x treating y as a constant.

    With fuller notation:

    You're working out \left(\dfrac{\partial \theta}{ \partial x}\right)_r

    But you actually want \left(\dfrac{\partial \theta}{ \partial x}\right)_y

    Hence the need to get r,theta in terms of x,y to start.
    Ahhh, thanks!
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    (Original post by ghostwalker)
    ]...
    Now I'm stuck on the second part.
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    I am extremely confused about limits.
    Attachment 524183524185
    Attached Images
     
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    (Original post by StarvingAutist)
    Now I'm stuck on the second part.
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Views: 54
Size:  3.0 KB
    I am extremely confused about limits.
    Attachment 524183524185
    Can we have the whole of the original question. Your graph doesn't match the equations.
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    (Original post by ghostwalker)
    Can we have the whole of the original question. Your graph doesn't match the equations.
    There's nothing more to it.

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    I changed my graph now, still no ideas.
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    (Original post by StarvingAutist)
    There's nothing more to it.

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    I changed my graph now, still no ideas.
    OK. For clarity, here's an image:

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    Now consider each of the 4 edges. What are their equations, and what do they become under the new coordinate system. The left/right edges are easiest, to start.
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    (Original post by ghostwalker)
    OK. For clarity, here's an image:

    Name:  Untitled.jpg
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Size:  12.7 KB

    Now consider each of the 4 edges. What are their equations, and what do they become under the new coordinate system. The left/right edges are easiest, to start.
    Got it in the end, not sure why I was confused. Anyway, does R = 0.25 (b^2 - a^2) * ln3 sound right?
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    (Original post by StarvingAutist)
    Got it in the end, not sure why I was confused. Anyway, does R = 0.25 (b^2 - a^2) * ln3 sound right?
    Not worked it through, but just graphed for b=5,a=3, and the area worked out to the value your formula gives.
 
 
 
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