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Math Help - differentiability multivariable function

  1. #1
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    differentiability multivariable function

    Hi, I need help on the following:

    Let  f : \mathbb{R}^2  \to \mathbb{R}^2 be defined by  f(x,y) = (y, x^2) . Prove that it is differentiable at  (0,0) .

    I know the partial derivatives exist at  (0,0) .

    Thanks
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  2. #2
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    Quote Originally Posted by storchfire1X View Post
    Hi, I need help on the following:

    Let  f : \mathbb{R}^2  \to \mathbb{R}^2 be defined by  f(x,y) = (y, x^2) . Prove that it is differentiable at  (0,0) .

    I know the partial derivatives exist at  (0,0) .

    Thanks
    So the derivative is going to be a 2 by 2 matrix.

    \begin{bmatrix} 0 & 1 \\ 2x & 0 \end{bmatrix}

    If you evaluate this at zero we get

    m=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}

    Now you need to calculate the limit

    f'(\mathbf{0})=\lim_{\mathbf{h} \to \mathbf{0}}\frac{||f(\mathbf{0}+\mathbf{h})-f(\mathbf{0})-m\mathbf{h}||}{||\mathbf{h}||}=0

    and show that it is equal to zero!

    Where

    \mathbf{h}=\begin{bmatrix}h_1 \\ h_2 \end{bmatrix}

    This should get you started.
    Last edited by TheEmptySet; May 9th 2011 at 08:51 PM. Reason: missing magnitude symbol
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by storchfire1X View Post
    Hi, I need help on the following:

    Let  f : \mathbb{R}^2  \to \mathbb{R}^2 be defined by  f(x,y) = (y, x^2) . Prove that it is differentiable at  (0,0) .

    I know the partial derivatives exist at  (0,0) .

    Thanks
    You can also use the fact that if \displaystyle \frac{\partial}{\partial x}f,\frac{\partial}{\partial y}f exist and are continuous on a neighborhood of (0,0) then f is differentiable there and f'(0,0)=\text{Jac}_f(0,0).
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