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Math Help - Area of a Surface

  1. #1
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    Area of a Surface

    Let \Phi(u,v) = (u-v,u+v,u) , and let D be the unit disk in the uv plane. Find the area of \Phi(D)

    I've parametrized u and v but i dont see how to put the angles in since it's asking for the area of the sphere.
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  2. #2
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    Quote Originally Posted by larryboi7 View Post
    Let \Phi(u,v) = (u-v,u+v,u) , and let D be the unit disk in the uv plane. Find the area of \Phi(D)

    I've parametrized u and v but i dont see how to put the angles in since it's asking for the area of the sphere.
    The Surface area is \displaystyle \iint_{D}\bigg| \bigg| \frac{\partial \Phi}{\partial u} \times \frac{\partial \Phi}{\partial v}\bigg| \bigg|dudv
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    The Surface area is \displaystyle \iint_{D}\bigg| \bigg| \frac{\partial \Phi}{\partial u} \times \frac{\partial \Phi}{\partial v}\bigg| \bigg|dudv
    Thank you but i already know the equation it's the setting it up part.
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  4. #4
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    You said "I've parametrized u and v" so I assume you have u= r cos(\theta), v= r sin(\theta). Put those in for u and v and integrate with r going from 0 to 1 and \theta from 0 to 2\pi in order to cover the unit circle.

    (No, it's not "asking for the area of the sphere". It is asking for the area of a region on the given plane.)
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  5. #5
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    It's a linear transformation

    T(\begin{bmatrix}u\\v \end{bmatrix})=\begin{bmatrix}u-v\\u+v\\u \end{bmatrix}\rightarrow T(\vec{e}_1)=\begin{bmatrix}1\\1\\1 \end{bmatrix}~,~<br />
T(\vec{e}_2)=\begin{bmatrix}-1\\1\\0 \end{bmatrix}

    A=\begin{bmatrix}1&-1\\1&1\\1&0 \end{bmatrix}~,~S_{\Phi}=\sqrt{\det (A^TA)}\cdot S_{D}=\sqrt{6}~\pi r^2=\sqrt{6}~\pi
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  6. #6
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    Thank You much appreciation.
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