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I would like to know one estimate of the euclidian norm of a derivative of a vector field composition. For example: Let $f:\mathbb R^n \to \mathbb R^m$ and $g:\mathbb R^p \to \mathbb R^n$ be two differentiable functions. What may I say about $||D(f\circ g)(x)||$? May I say that $\|D(f\circ g)(x)\| \le\|Df(g(x))\||\mathrm{jac}\,g(x)|$?

Thanks in advance

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    The chain rule converts this into a question about two rectangular matrices $A$ and $B$: what can we say about $\|AB\|$? Well, $\|AB\|\le \|A\|\|B\|$. To get anything more, you need to know something extra about these matrices. The fact that they arise as derivatives tells us nothing: any matrix is the derivative of the corresponding linear map.2012-06-20

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