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Let $ \mathbf{ G}: \Bbb R^2 \to \Bbb R^2 $ be given by $G(x,y)=(cos(x-y),sin(x+y)) $

Assume that $f:\Bbb R^2 \to \Bbb R$ is differentiable, with gradient $\nabla f(0,1) = (2,3)$ in $(0,1)$. Let $h: \Bbb R^2 \to \Bbb R$ be given by $h(x)=f(\mathbf{G}(\mathbf{x}))$ for all $x$.

What is the gradient $\nabla h( \frac{\pi}{2},0 )$ of $h$ in $(\frac{\pi}{2},0)$?

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hint: use the chain rule – Quickbeam2k1 Mar 12 '13 at 18:46
up vote 1 down vote accepted

Since $G(\pi/2,0)=(0,1)$ you get: $$ \nabla h = D f(0,1)\cdot DG(\pi/2,0) $$ which is a $1\times 2$ times $2\times 2$ matrix multiplication.

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