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Thread: Jacobian of a composed function+chain rule

  1. #1
    MHF Contributor arbolis's Avatar
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    Jacobian of a composed function+chain rule

    Consider the following functions : $\displaystyle f\begin{pmatrix} u \\ v \end{pmatrix}=\begin{pmatrix} u+v \\ u-v \\ u^2-v^2 \end{pmatrix}$ and $\displaystyle F(x,y,z)=x^2+y^2+z^2=w$.
    1)Find the Jacobian of $\displaystyle F \circ f$ in $\displaystyle f\begin{pmatrix} a \\ b \end{pmatrix}$.
    2)Find $\displaystyle \frac{\partial w}{\partial u}$ and $\displaystyle \frac{\partial w}{\partial v}$.

    My attempt :
    1) $\displaystyle (F \circ f)(a,b)=(a+b)^2+(a-b)^2+(a^2-b^2)^2$. And the Jacobian is $\displaystyle \left ( 2a(2+2a^2-b^2) \hspace{1 cm} 2b(2+2b^2-a^2) \right )$. Is it right?

    2)$\displaystyle \frac{\partial w}{\partial u}=\frac{\partial w}{\partial x}\cdot \frac{\partial x}{\partial u}+\frac{\partial w}{\partial y}\cdot \frac{\partial y}{\partial u}+\frac{\partial w}{\partial z}\cdot \frac{\partial z}{\partial u}$. But I'm stuck on how to find $\displaystyle \frac{\partial x}{\partial u}$ for example. Any idea?
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  2. #2
    Moo
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    Hello,

    For question 2), what I don't see is how w and u can be related, as they're just dummy variables (that is to say : their name is not important)
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    For question 2), what I don't see is how w and u can be related, as they're just dummy variables (that is to say : their name is not important)
    Unfortunately the question is as I posted here. I guess I have to use the Jacobian I calculated in part 1), but I don't see how.
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