0
votes
0answers
20 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
1
vote
0answers
34 views

Integral of a determinant of Jacobian depends on the boundary values only

Let $B$ be the closed unit ball in $\mathrm{R}^n$ with the 2-norm. Let $\phi : B \to \mathrm{R}^n$ be smooth such that $\det D \phi = 1$ on $\partial B$. Why is $\int_B \det D \phi = \int_B 1$? In ...
2
votes
1answer
186 views

Line integral using variable change

The variable change theorem is the following: $$\int_B f = \int_A f \circ g \cdot |det\mathcal Jg|$$ So to calculate the following line integral: $$\int_C(xy)ds$$ where $C = g(t) = (cost, ...
0
votes
1answer
57 views

Proof of the naturality of integration

I have a bit of a problem with the following identity: Suppose that $U, V \subset \mathbb{R}^n$, are two open sets. Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
11
votes
2answers
399 views

Integral of determinant

Good evening. I need help with this task $$ \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta ...