All Questions
56
questions
2
votes
1
answer
203
views
Source of the definition of integrating a form along a curve in a manifold
Suppose that $M$ is a smooth manifold. Let $\omega$ be an $n-$ form on $M$ with compact support. Then we define $\int_M\omega$ using partitions of unity. If $M$ is covered by a single chart $h:M\to \...
- 11.5k
0
votes
1
answer
39
views
Shell theorem of an oblate spheroid.
Isaac Newton famously proved the shell theorem and stated that:
"A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at ...
- 1,111
-1
votes
1
answer
42
views
Integral of diffeomorphism [closed]
I'm studying differential geometry, and there is this passage I don't understand: we have a diffeomorphism f of $R^k$ such that $f(0)=0$. Then he says that $f(x)=\int_{0}^{1}\frac{d}{ds} f(xs) ds $ ...
- 3
0
votes
0
answers
33
views
Time derivative of integrals on changing level and upper contour sets
There are related questions and answers on StackExchange but I cannot find a general formula for the following types of (time) derivatives of integrals with respect to changing level and upper contour ...
- 93
1
vote
0
answers
48
views
Is indepedence of pair of functions equivalent to pointwise orthogonal gradients?
Let U be a domain in $\mathbb{R}^n$ and $f_1$, $f_2$: $U \to \mathbb{R}$ differentiable functions on $U$. Let $h_1$, $h_2$ be two "arbitrary" functions from $\mathbb{R} \to \mathbb{R}$ (...
- 421
0
votes
1
answer
149
views
Area of a rectangle in a Poincare half-plane
Let (-1,1), (-1,2), (1,1), and (1,2) be points in the Poincaré half-plane and consider the rectangle formed by joining the points with geodesics. I want to find the area of this rectangle using ...
- 1
2
votes
2
answers
163
views
What is the "polar coordinate function"?
I'm reading Tapp's Differential Geometry of Curves and Surfaces, I have this problem:
$\quad\color{green}{\text{Exercise 1.10.}}$ Prove that the arc length, $L$, of the graph of the polar coordinate ...
- 24.2k
3
votes
1
answer
351
views
Calculating a line integral using Green's theorem in a region with a singularity
Problem:
Calculate the line integral
$$
\int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1}
$$
where $A$ is the line $|x|+|y|=4$, travelling clockwise and making one rotation.
Answer: $-2\pi$
Solution:
The ...
- 89
18
votes
0
answers
2k
views
A difficult integral for the Chern number
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +(m-\cos x-\cos y)^2\right)^{3/2}}
$$
gives the ...
- 335
0
votes
0
answers
63
views
Reference request: Calculus 3 from Differential Geometry perspective
I am looking for a book (or chapter, section, etc.) that explains multivariable integration, line integrals and other "Calculus 3" (so-called in the USA) integration topics from a ...
- 61
1
vote
1
answer
204
views
Find the centroid of a symmetric compact manifold.
What is shown below is a reference from the text Analysis on Manifolds by James Munkres
So we say that the centroid $C_M$ of a compact $k$-manifold $M$ of $\Bbb R^n$ is the point of $\Bbb R^n$ whose $...
- 4,906
1
vote
1
answer
50
views
Using the surface of the unit sphere to calculate $\int_{\mathbb R^3}\chi_{\{x_3>0\}}\chi_{B^c_{1/n}(0)}\exp(-\lvert x\rvert^2)/\lvert x\rvert dx$
I'm struggling with finding a proof idea for this integral from an integration theory practice exam.
Suppose
$$f_n:=\frac{\exp(-\lVert x\rVert_2^2)}{\lVert x\rVert_2}\chi_{\{x_3>0\}}(x)\chi_{B^c_{1/...
- 2,276
1
vote
0
answers
58
views
Verifying Gauss's Theorem
Let
$B:=\{(x,y): |x| \leq 3, |y| \leq 3, x^2+y^2 \geq 1\}$ and the vector field $v(x,y)={(x-y,x)}^{T}$
I need to draw B and the boundary $\partial B$.
I drew the region. It would be the red lined part....
- 547
1
vote
1
answer
229
views
Prove that the integral of a differential form on an open set is well defined.
What to follow is a summary of part of the sixth chapter of the text Analysis on Manifolds by James Munkres.
Definition
Given $x\in\Bbb R^n$ we define a tangent vector to $\Bbb R^n$ at $x$ to be a ...
- 4,906
3
votes
0
answers
825
views
How would I find the volume of a torus with an elliptical cross-section using calculus?
I know that, by Pappus' centroid theorem, the volume of a torus with an elliptic cross-section of major axis $a$ and minor axis $b$ is $2𝜋^2abc$, where $c$ is the distance from the center of the ...
- 31