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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 $f:M\to\mathbb ... 0answers 34 views ### Integral of a determinant of Jacobian depends on the boundary values only Let$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 ... 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, ...
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 $V$. ...
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 ...