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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. –  user31373 Jun 20 '12 at 15:02
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This statement is false : let $f(x,y)=(x,y)$ and $g(x,y)=(x+y,x+y)$, then $f\circ g=g$ and $D(f\circ g)(x)\neq 0$, so $\|D(f\circ g)(x)\|\neq 0$. But $|\mathrm{Jac}\,g(x)|=0$. So you can't have $\|D(f\circ g)(x)\| \le\|Df(g(x))\||\mathrm{jac}\,g(x)|$.

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The answer is no. Consider the following functions $f:R^2 \to R$ and $g: R^2 \to R^2$ defined by $$f(x,y)= x^2+y^2 \quad g(x,y)= (x+2y, x+2y). $$ We verify that $$\|Df(g(x,y))\|>0 \quad \textrm{ and } \quad \textrm{jac}\ g(x,y)=0.$$ –  jalmanalves Jun 20 '12 at 14:17
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