Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am told that when transforming coordinates $\nabla (f(Ax)) = A^T(\nabla f)(Ax)$, however I read both of these as "the gradient of f, where f is a function of Ax, where A is a matrix and x is a vector", with the second multiplied with $A^T$.

This is obviously incorrect, so how am I misunderstanding the notation?

Thanks, Ash

share|improve this question
add comment

1 Answer

up vote 5 down vote accepted

There can be a bit of subtlety whenever the gradient operator appears, because it is a differential operator, but the notation doesn't always make it clear what variable the differentiation is with respect to.

On the left, you are differentiating $f(Ax)$ with respect to $x$. This is not the gradient of $f$ per se, but of the transformed function $x \mapsto f(Ax)$. On the right, $\nabla f$ is the gradient of $f$. So $(\nabla f)(Ax)$ is the gradient of $f$ evaluated at $Ax$; in other words, you are differentiating $f$ with respect to its argument (which here is $Ax$), not with respect to $x$.

It might help to think of the analogous one-dimensional equation: $\frac{d}{dx} f(ax) = a f'(ax)$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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