Jacobian of a composed function+chain rule

**arbolis** · July 26th 2009, 09:22 PM

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

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

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

**Moo** · July 26th 2009, 11:29 PM

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)

**arbolis** · July 27th 2009, 09:37 AM

Originally Posted by Moo

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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