All Questions
Tagged with integration multivariable-calculus
1,196
questions with no upvoted or accepted answers
24
votes
0
answers
796
views
When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?
Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
- 11.7k
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
13
votes
0
answers
181
views
How weird can the boundary be so that the fundamental theorems of vector calculus hold?
Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
- 15.3k
9
votes
0
answers
904
views
Nasty Integral - Closed form solution?
Any suggestions on how to integrate this beast?:
$$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2\left(\frac{\omega_{12}}{2}\right)\sin^2\left(\frac{\omega_{23}}{2}\right)d\omega_{23}d\omega_{12}$$
where:
...
- 2,509
8
votes
1
answer
994
views
Surface integral for a scalar function defined on a discrete surface
Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
- 433
7
votes
0
answers
197
views
Understanding intuition behind integral involving mixed product
The topological charge i.e. skyrmion number (also called wrapping number) is defined as the number of times the spin vectors in a 2D configuration (i.e. lying on a 2D plane, as shown in the image ...
- 346
7
votes
0
answers
318
views
When is a surface integral equal to double integral over projection? A verification of Stokes' Theorem. Intuition and relation to Green's Theorem.
I'm helping a calculus student. It's been awhile since I've done some vector calculus. I know Stokes' Theorem in terms of differential forms and manifolds with boundary, but I've forgotten much of ...
user636310
6
votes
0
answers
296
views
Rudin's PMA 10.38 theorem
Suppose $E$ is a convex open set in $\Bbb R^{n}$, $f \in C^{1}$, $p$ is an integer, $1\leq p \leq n$ and $(D_{j}f)(x)=0$ ($p<j \leq n ,x\in E$).
Then there exists an $F \in C^{1}$ such that $...
- 1,020
6
votes
1
answer
3k
views
Find the area of the part of the paraboloid $z = x^2 + y^2$ that lies under the plane $z=4-x$
So the first thing I did was to try and parametrize the parabloid as:
$$r(\theta,z)=\sqrt{z}\cos(\theta)i+\sqrt{z}\sin(\theta)j+zk$$
Then I found $||r_\theta\times r_z||=\frac{1}{2}\sqrt{4z+1}$.
...
- 2,457
6
votes
0
answers
284
views
How to prove following integral equality?
Let's have the equality
$$
\int \limits_{-\infty}^{\infty} \left[ [\nabla_{\mathbf r'} \times \mathbf A (\mathbf r' )] \times \frac{\mathbf r' - \mathbf r}{|\mathbf r' - \mathbf r |^{3}}\right]d^{3}\...
6
votes
0
answers
319
views
Algorithm to calculate multiple integral.
One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
- 14.8k
5
votes
0
answers
60
views
Are there programs which compute integrals in $\mathbb{R}^n$?
I look for a software which computes integrals like, for example, this one:
$$\int_{B_R}\frac{|x|^3 dx}{(\varepsilon^2+|x|^2)^{n-1}},$$
where $\varepsilon>0$ and $R>0$ are not specific numbers ...
5
votes
0
answers
77
views
Does $ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x+y)^{1000}}\ dx\,dy $ converge?
I want to check if the following integral diverges or converges.
$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x+y)^{1000}}\ dx\,dy $
I'm not sure if it diverges or converges. I attempted to ...
- 391
5
votes
0
answers
133
views
Show that surface integral of absolute vorticity is constant over time
In a rotating frame the (unforced, incompressible) Euler equation is
$$
\frac{∂\vec{u}}{∂t}+\vec{u}\cdot\nabla\vec{u}=-\nabla\left(\frac{p}{\rho_0}\right)-2\vec{\Omega}\times\vec{u}+\nabla\left(\frac{...
- 284
5
votes
0
answers
869
views
Evans Partial Differential Equations, chapter 7 exercise 1 (uniqueness of a regular solution to the heat equation with Neumann boundary conditions)
I would like to know how to solve the first exercise of chapter 7 of Evans' Partial Differential Equations, second edition. The problem goes like this:
Let $U\subset\mathbb{R}^n$ be an open and ...
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