All Questions
Tagged with integration multivariable-calculus
5,463
questions
77
votes
2
answers
9k
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Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite ...
user9464
71
votes
2
answers
27k
views
Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?
If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is:
$$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt.
\qquad\text{(1)}$$
If $T=\{x=f(u,v); y=g(u,v)\}$ ...
- 821
67
votes
2
answers
3k
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Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$
Here is my source of inspiration for this question.
I suggest to evaluate the following new one.
$$
I_{n}:= \int_0^1 \! \cdots \! \int_0^1
\left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}...
- 121k
53
votes
6
answers
7k
views
The Meaning of the Fundamental Theorem of Calculus
I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
- 2,909
47
votes
4
answers
10k
views
Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\...
- 5,527
46
votes
5
answers
19k
views
Interpreting Line Integrals with respect to $x$ or $y$
A line integral (with respect to arc length) can be interpreted geometrically as the area under $f(x,y)$ along $C$ as in the picture. You sum up the areas of all the infinitesimally small 'rectangles' ...
- 5,361
44
votes
3
answers
9k
views
How to change variables in a surface integral without parametrizing
This is a doubt that I carry since my PDE classes.
Some background (skippable):
In the multivariable calculus course at my university we made all sorts of standard calculations involving surface ...
- 3,447
33
votes
4
answers
20k
views
Proof of the Gauss-Green Theorem
I can't seem to find any references that gives a proof of the Gauss-Green theorem:
Let $U\subset\mathbb{R}^{n}$ be an open, bounded set with $\partial U$ being $C^1$. Suppose $u\in C^{1}(\bar{U})$, ...
- 4,305
24
votes
2
answers
40k
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When integrating how do I choose wisely between Green's, Stokes' and Divergence?
I've been taught Green's Theorem, Stokes' Theorem and the Divergence Theorem, but I don't understand them very well. In particular I don't understand in what circumstances I would choose to use any ...
- 919
24
votes
1
answer
776
views
Tricky surface integral of vector field
The following problem comes from a vector calculus exam.
Let $$ S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\} $$ be an embedded surface with the orientation ...
- 1,004
24
votes
0
answers
796
views
When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?
Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
- 11.7k
23
votes
3
answers
2k
views
distance from the centre of a $n$-cube as $n \rightarrow \infty$
I've figured out the pattern for calculating the average distance from the centre of an n-cube; but I don't have a formula for the answer. Is there an easy way to figure this out?
Average distance of ...
- 2,554
22
votes
4
answers
953
views
Evaluating $\int_0^1\int_0^1\cdots\int_0^1\frac{n\max\{x_1,x_2,\cdots,x_n\}}{x_1+x_2+\cdots+x_n}dx_1dx_2\cdots dx_n$
I was trying to compute the following integral:
$$I=\int_0^1\int_0^1\cdots \int_0^1 \frac{n \max\{x_1,x_2,\ldots,x_n\}}{x_1+x_2+\ldots+x_n}dx_1dx_2\ldots dx_n.$$
My attempt was: Let $X_1,X_2, \ldots, ...
- 1,121
22
votes
3
answers
18k
views
Volume integral of the curl of a vector field
I am having hard time recalling some of the theorems of vector calculus. I want to calculate the volume integral of the curl of a vector field, which would give a vector as the answer. Is there any ...
user23238
22
votes
4
answers
890
views
Help with Seemingly Hopeless Double Integral
I hate to be that guy to just post an integration problem and ask how to solve it so I'll give a little relevant info
Okay, so I'm working on a physics project and my professor proposed that the ...
- 4,258