All Questions
Tagged with integration multivariable-calculus
5,463
questions
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2
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Is $\int_D \nabla f dx = 0$ for $f$ compactly supported?
Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in
$$\int_D \nabla f dx.$$
If $n = 1$, then by the ...
- 6,277
2
votes
0
answers
148
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Calculate the Integral $\iint x^{2}\:dydz + (x^2 + y^2 + z^2 )\:dx dy$ [closed]
The integral that I need to calculate is the integral of the second kind:
$$I=\iint_{S} x^{2}\:dydz + (x^2 + y^2 + z^2 )\:dx dy$$
with $S$ being the boundary of the following object: $x^2 + y^2 \leq z ...
- 480
3
votes
0
answers
57
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Calculation of the volume of a solid of rotation
Calculate the centroid of the homogeneous solid generated by the rotation around the y-axis of the domain in the xy-plane defined by: $$D=\left \{(x,y)\in \Bbb R^2 : x \in [1,2],0\leq y \leq \frac{1}{...
2
votes
2
answers
76
views
How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
- 23
1
vote
1
answer
46
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Trouble with 3D Fourier transform of a cross product expression
I don't understand a Fourier transform identity that has been quoted with no source in several papers (relevant to quantum mechanics):
$$
\vec f(\vec r_i)=\int \frac{\vec r_i - \vec r_j}{|\vec r_i - \...
- 13
0
votes
1
answer
121
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A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz
The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being
$$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
- 3,679
2
votes
1
answer
67
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Evaluation of the given line integral
Question: Evaluate $\int_{C}$B.dr along the curve $x^{2}$+$y^{2}$=1,$z$= 1 in the positive direction from (0,1,2) to (1,0,2);given
B= (xz²+y)i+(z-y)j+(xy-z)k
The question itself is easy,but I don't ...
- 33
7
votes
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answers
197
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Understanding intuition behind integral involving mixed product
The topological charge i.e. skyrmion number (also called wrapping number) is defined as the number of times the spin vectors in a 2D configuration (i.e. lying on a 2D plane, as shown in the image ...
- 346
3
votes
1
answer
115
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Computing an integral using differential under the integral sign
The following integral is in question.
$$I(x) =\int_0^x \frac{\ln(1+tx)}{1+t^2}\,dt$$
My attempt is finding $I’(x)$ which is
$$I’(x) = \int_0^x \frac{t}{(1+t^2)(1+tx)}\,dt + \frac{\ln(1+x^2)}{1+x^2} $$...
- 617
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votes
1
answer
36
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Parameterising Surfaces Integration
$S$ is the surface of a cube which is bounded by $6$ planes being $x=1,x=3,y=2,y=4,z=0,z=2$. The normal vector points outwards and the vector field is $F = (x^2-sin(yz),\frac{cos(x)}{x^2}-yz,x^2y)$
...
- 261
1
vote
1
answer
37
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Changing the integration limits of a triple integral
I have a triple integral of the form
$$
\int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_1} dt_3\ f(t_1,t_2,t_3)
$$
and I want to transform it to the form
$$
\int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3\...
- 451
3
votes
1
answer
128
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how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$
How to evaluate: \begin{align*}
&\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
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2
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70
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1
vote
2
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54
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Prove that $ \int_{\Omega} \Vert \nabla u\Vert^2 -6F(u)dx=-\int_{\partial \Omega} \Vert \nabla u\Vert^2 x\cdot \vec{n}d\sigma$
Let $\Omega\in \mathbb{R}^3$ be a compact region, $f \in C(\mathbb{R})$ and $F(r)=\int_{0}^{r}f(t) dt$. If $u\in C^2(\Omega)$ satisfies $\Delta u(x)+f(u(x))=0,x\in \Omega$ and $u(x)=0,x\in \partial \...
- 51
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0
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21
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Interchanging Bounds in Double Integrals (When Bounds are Functions of y or x)
Let me set the scene, we have a domain of a function of two variables $f(x,y)$.
$$D = \{(x,y): a \leq x \leq b, g_1(y) \leq y \leq g_2(y)\}$$
So on the $y$-axis it's bounded by two functions. So when ...
