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 $ax+b$?


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 $$ x \mapsto (x,ax+b) \mapsto \left(\cos x , \cos (ax+b), \sin(ax+b)\right). $$

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.