All Questions
529
questions
0
votes
1
answer
48
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
- 174
-4
votes
2
answers
152
views
Theorem 16.5, Munkres' Analysis on Manifolds [closed]
In Munkres' Analysis on Manifolds, page 142 Theorem 16.5 it states:
$$\int_{D}f\leq\int_{A}f$$
at the end of that page.
Here, $D=S_{1}\cup\cdots\cup S_{N}$ is compact since $S_{i}=Support\phi_{i}$ and ...
- 174
1
vote
0
answers
61
views
A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
- 3,065
0
votes
1
answer
62
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$.
Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
- 5,450
0
votes
1
answer
77
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
2
votes
1
answer
67
views
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
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
$$
\int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3\...
- 451
1
vote
0
answers
46
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A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists
A mistake in Munkres' Analysis on Manifolds about proving if $D$ has measure zero in $\mathbf{R}^{n}$, then $\int_{Q}f$ exists
In Munkres' Analysis on Manifolds, page 94 Theorem 11.2 it states: Divide ...
- 174
5
votes
1
answer
136
views
General formula for reversing double integral bounds
The double integral over the region:
$$
R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\}
$$
is expressed as
$$
\...
- 170
1
vote
1
answer
61
views
Integrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral:
$$
\operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{\,\,n}}\,
\left\vert\rule{0pt}{4mm}\,{\cos\...
- 888
0
votes
2
answers
62
views
How to calculate a multiple integral over a triangular region
I am having trouble computing the multiple integral
\begin{equation}
\int_0^1 \int_0^{1-x} e^{\frac{1}{2}(x + y)^2} \, dy\, dx
\end{equation}
Because integrating $e^{\frac{1}{2}(x + y)^2}$ with ...
0
votes
0
answers
45
views
Calculating the volume of a body given by $g(x,y,z) = xyz$ using triple integrals
I need help calculating the volume of the region/body/solid given by
$D = \{(x,y,z) \colon 0 \leq xyz \leq 8 \text{ and } 0 \leq x \leq 2, 0 \leq y \leq 2, 0 \leq z \leq 2\}$,
I am supposed to do it ...
- 31
1
vote
1
answer
147
views
Triple Schwinger integral
I'm working on a perturbative QFT problem which requires three Schwinger parametrizations; that is,
$$
\frac{1}{A^n}=\frac{1}{\Gamma(n)}\int_0^\infty dz\ e^{-A z}z^{n-1}.
$$
In the end, after taking ...
- 89
4
votes
0
answers
53
views
evaluate the volume of solid
Consider the paraboloid $(\mathcal{P}): z=x^2+y^2$ and the plane $(\mathcal{Q}): 2x+2y+z=2$.
Let $\mathcal{S}$ be the solid region bounded above by $(\mathcal{Q})$ and below by $(\mathcal{P})$. Find ...
- 319
0
votes
2
answers
39
views
set the limits of integration of the spherical coordinates between two paraboloids and a plane
Find the volume of the solid $\mathcal{S}$ enclosed laterally by the paraboloids $\mathcal{P}_1$ of equation $z = x^2 + y^2$ and $\mathcal{P}_2$ of equation $z = 3(x^2 + y^2)$ and from above by the ...
- 319