Pullback of a Volume Form Under a Diffeomorphism.

Question

I have an exercise here, which I have no idea how to do.

Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that $$ {f^{*}}(\text{vol}_{V}) = \sqrt{\det \! \left( \left[ \left\langle {\partial_{i} f}(\bullet),{\partial_{j} f}(\bullet) \right\rangle \right]_{i,j = 1}^{n} \right)} \cdot \text{vol}_{U}. $$ Notation:

• $ \text{vol}_{U} $ and $ \text{vol}_{V} $ denote the volume forms on $ U $ and $ V $ respectively.
• $ f^{*}: {\Lambda^{n}}(T^{*} V) \to {\Lambda^{n}}(T^{*} U) $ denotes the pullback operation on differential $ n $-forms corresponding to $ f $.

Answer 1

I don't have enough for a comment (my original account was wiped out), but this is my impression:I think this is just the change-of-basis theorem/result; if $U,V$ are open balls, a diffeomorphism is basically a coordinate change map, and so it transforms according to the (determinant of the ) Jacobian.

Answer 2

Hint: I believe you have a typo. This should be be $f_i\cdot f_j$. Here the subscript denotes partial derivative. Now look up the Gram determinant.