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Suppose I have the map $R^2\rightarrow R^3, (x,y)\mapsto (\cos x, \cos y, \sin y)$ and I want to compute the Jacobian matrix of the restriction of this map to $y=ax+b$, i.e., $(x,ax+b)\mapsto (\cos x, \cos (ax+b), \sin (ax+b))$. How do I do it? How to differentiate with respect to the variable $ax+b$?

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Just compute the composition of the two maps.

The restriction is given by the map $\mathbb R \to \mathbb R^2$ defined by $x \mapsto (x,ax+b)$.

Then the composition is given by $$ x \mapsto (x,ax+b) \mapsto \left(\cos x , \cos (ax+b), \sin(ax+b)\right). $$

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  • $\begingroup$ So for example the result of differentiating $\cos(ax+b)$ w.r.t. $ax+b$ is $-a\sin(ax+b)$? $\endgroup$ – user437309 Jan 31 '18 at 19:17
  • $\begingroup$ You cannot differentiate with respect to an expression. You differentiate with respect to a variable. $\endgroup$ – Fredrik Meyer Jan 31 '18 at 20:31

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