All Questions
Tagged with integration multivariable-calculus
5,463
questions
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Simplification of Dependent Nested Integrals
I have been working with nested Integrals of the following form :
$$ \int_0^{T}dt_n \int_0^{t_n}dt_{n-1} \cdot \cdot \cdot \int_0^{t_2} dt_1 \prod_{i=1}^n f_i(t_i) $$
Some simpler cases and prework :
...
- 11
0
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1
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91
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Variable transformation in the definite integral
recently I encounter a variable transformation problem in the derivation and I did not figure out how it works.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
- 41
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0
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69
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Looking for citation of a definition of multivariable integral
I need a definition of Multivariable Riemann Integral to cite in my article.
I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books.
In the first two I ...
0
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0
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47
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Am I correctly applying repeated integration by parts?
Say we have two compactly supported functions $f,g:\mathbb{R}^n\to\mathbb{R}$. I found myself computing
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n}f\frac{\partial^{r}g}{\partial x^{\alpha_1}\...
- 5,166
-1
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0
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19
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Need help with the steps and limits in this multivariable integration of a joint probability density function
I am not sure how to proceed with this double integration. I know this can be evaluated in a much easier way than solving the integral as its just the volume of a cube but I need help with the process ...
0
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0
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37
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Intergrating $ \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $
I want to integrate $$ I = \int_{M} f (x.,y,z,w)\ d {\rm
Vol}_3 $$where $f(x,y,z,w) = (x+y)e^{z+w} $, and
$M = \{x+y+z+w = 1, x,y,z,w > 0\} $.
I need to find a parameterization of M; if I consider $...
- 391
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1
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48
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
- 174
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2
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66
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Integration by parts on an area
I'm reading an Engineering book. All I can think of is integration by parts
$$\int_{\Omega} \dfrac{\partial M_y}{\partial x}v\text{d}\Omega = M_y v|_{?}^{?} - \int_{\Omega} M_y\dfrac{\partial v}{\...
- 171
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24
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Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function
I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
- 5,064
5
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1
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Integral calculated directly and with Gauss Green formula
I checked on Wolfram Alpha that the directly calculated integral was correct, when I go to use the Gauss Green formula, I get $0$ as a result, when instead, I ...
- 179
-4
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2
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152
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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
3
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1
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86
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Integral in polar coordinates
I was wondering if I had set up this integral correctly. If anyone could help me, I would greatly appreciate it. I am available in case there are any unclear things!
$$\iint_D x^2+y^2dxdy, \quad D=\{1\...
- 179
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61
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A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
- 3,065
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1
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Evaluate $\iint_S\vec{F}$ where $S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above and below by two planes.
Use the divergence theorem to evaluate $$\iint_S\vec{F}$$ where $\vec F(x,y,z)=\langle xy^2,x^2y,y \rangle$ and $\vec S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above ...
- 3,065
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0
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54
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How to analyze the behavior of this function?
I have a map from positive reals to positive reals, of the form $$ f(d_1) = \frac{\int_0^\pi \int_0^\pi \int_0^{2\pi} 4\pi sin^2(\frac{\theta}{2})(1+(1-cos\theta )(v_1^2-1)) e^{k(d_1 + d_2 + d_3 + (1-...
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1
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69
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
I tried solving this problem as follows:
Equation of the cylinder $x^2+(y-a)^2=...
- 3,065
3
votes
1
answer
61
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
I tried solving this problem as follows:
The equation ...
- 3,065
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1
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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
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2
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79
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What does the integral $\int_0^V d\vec{x}$ mean/represent and how is it calculated?
My question is what does an integral such as
$$\int_0^V d\vec{x}_1$$
mean exactly?
Here is the context in which this type of integral arose.
I am following a thermodynamics course that has a section ...
- 5,021
0
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0
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29
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integration by parts $\int_{\mathbb{R}^N} -\Delta u (u-M)\varphi dx$
I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. ...
- 89
1
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1
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283
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Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277
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0
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20
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How to integrate function involving gradient of multivariate function?
Let's say we have a function f(x,y) = $x^2 + y^2$ , where x and y are real numbers and I have to compute the integral
$ I = \int g(v) \triangledown_{v} f(v) dv $, where v = [x, y]$^T$ and g(v) is a $R^...
- 23
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1
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77
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$
I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
5
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1
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158
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Calculate the volume of intersection sphere and cone using triple integral
Compute the volume under sphere $x^2+y^2+z^2=4$ above $xy$-plane and above cone $z=\sqrt{x^2+y^2}$ using triple integral.
I try to plot as follows:
I try using triple integral
$$\int\limits_{-\sqrt{2}...
- 2,354
1
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1
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77
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Change the order of integration: $\int_0^2\int_y^{2y}xy dxdy$
I was presented on a class today the following integral and asked to calculate it and then try to change the order of integration: $$\int_0^2\int_y^{2y}xydxdy$$
As the integral is pretty simple I didn'...
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1
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90
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How to solve the integral $\iint_D(x^2+y^2)^{-2} dxdy$ with where $D$ is $x^2+y^2\leq2, x\geq 1$ and after that the same integral but with $x\leq1$. [closed]
So my main question is after I use the Jacobian how do I write the new space $D^*$ when I substitute $x=r\cos θ$ and $y=r\sin θ$. I know that I will find what the range of $θ$ is by simply finding the ...
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37
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How to solve the $\iint_D \frac{(x-y)^2}{1+x+y}dxdy$ where D is the trapezoid with edges $(1,0),(0,1),(2,0),(0,2)$ and using $u=1+x+y$ and $v=x-y$
I know that you need to find the Jacobian of $u,v$ and multiply it with the existing $f(x,y)$ inside the integral. But how can I then find the limits of integration for $u$ and $v$ after that since ...
3
votes
1
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64
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Find volume of body between surfaces
Problem: Find volume of body defined as follows:
$z^2=xy$, $(\frac{x^2}{2}+\frac{y^2}{3})^4=\frac{xy}{\sqrt{6}}$, $x, y, z \ge 0$.
My solution:
So we're working in the all positive octant of the ...
- 41
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3
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134
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Find area of figure and volume of body
Problem
Find area:
$$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2=\frac{xy}{c^2},\qquad a,b,c > 0$$
My solution:
The figure encloses area under itself so we're looking at: $$\left(\frac{x^2}{a^2}+...
- 65
2
votes
1
answer
45
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region above the cone and inside the sphere
I've come across a problem where
\begin{array}{c}
\mbox{I have to compute the volume of the region}
\\ \mbox{above the cone}\
z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1
\...
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2
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66
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Is $\int_D \nabla f dx = 0$ for $f$ compactly supported?
Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in
$$\int_D \nabla f dx.$$
If $n = 1$, then by the ...
- 6,277
2
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0
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148
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Calculate the Integral $\iint x^{2}\:dydz + (x^2 + y^2 + z^2 )\:dx dy$ [closed]
The integral that I need to calculate is the integral of the second kind:
$$I=\iint_{S} x^{2}\:dydz + (x^2 + y^2 + z^2 )\:dx dy$$
with $S$ being the boundary of the following object: $x^2 + y^2 \leq z ...
- 480
3
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0
answers
57
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Calculation of the volume of a solid of rotation
Calculate the centroid of the homogeneous solid generated by the rotation around the y-axis of the domain in the xy-plane defined by: $$D=\left \{(x,y)\in \Bbb R^2 : x \in [1,2],0\leq y \leq \frac{1}{...
2
votes
2
answers
76
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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
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1
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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):
$$
\vec f(\vec r_i)=\int \frac{\vec r_i - \vec r_j}{|\vec r_i - \...
- 13
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1
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121
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A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz
The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being
$$ \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
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1
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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
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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
$$
\int_0^t dt_1 \int_0^{t_1} dt_2 \int_0^{t_2} dt_3\...
- 451
3
votes
1
answer
128
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how to evaluate $\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{\sqrt{xy}(1+xy) \log_{\pi}^{2}{xy}} \, dx \, dy$
How to evaluate: \begin{align*}
&\int_{0}^{1}\int_{0}^{1} \frac{2-x-y}{(\sqrt{xy}+\sqrt{x^{3} y^{3}}) \left[ \log_{\pi}^{2}{x} + 4\log_{\pi}{\sqrt{x}} \log_{\pi}{y} + \log_{\pi}^{2}{y} \right]} \, ...
0
votes
2
answers
70
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1
vote
2
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54
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Prove that $ \int_{\Omega} \Vert \nabla u\Vert^2 -6F(u)dx=-\int_{\partial \Omega} \Vert \nabla u\Vert^2 x\cdot \vec{n}d\sigma$
Let $\Omega\in \mathbb{R}^3$ be a compact region, $f \in C(\mathbb{R})$ and $F(r)=\int_{0}^{r}f(t) dt$. If $u\in C^2(\Omega)$ satisfies $\Delta u(x)+f(u(x))=0,x\in \Omega$ and $u(x)=0,x\in \partial \...
- 51
0
votes
0
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21
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Interchanging Bounds in Double Integrals (When Bounds are Functions of y or x)
Let me set the scene, we have a domain of a function of two variables $f(x,y)$.
$$D = \{(x,y): a \leq x \leq b, g_1(y) \leq y \leq g_2(y)\}$$
So on the $y$-axis it's bounded by two functions. So when ...
2
votes
1
answer
32
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Expressing integral based on the change of variables in double integral
Let B be the region in the first quadrant bounded by the curves $xy=1$,$xy=3$,$x^2−y^2=1$, and $x^2−y^2=4.$
Evaluate $\iint_{B} (x^2 + y^2 ) dx dy$ using the change of
variables $u=x^2−y^2$,$v=xy.$
...
0
votes
0
answers
29
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How can we calculate the Integral of a vector norm along a parametrized line?
I feel a bit silly because I've been trying to calculate the following integral and I believe that the solution is likely extremely easy, but I am blanking. Let $p \in (0,1)$, Given two vectors $\xi$ ...
0
votes
0
answers
37
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Is this calculus derivation process correct?
This is part of the economics romer model, finding the Lagrangian value
$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\
s.t.\int_{i=0}^{A}L(i)^{\phi}di=1
$...
- 101
3
votes
1
answer
323
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Flux calculation - what did I do wrong?
The exercise asks to calculate the flux of $\mathbf{F}=(4x,4y,z^2)$ through the surface $x^2+y^2=25, 0\le z \le 2$
I calculated using Gauss' theorem and I obtained $500\pi$, which is, as the teacher ...
- 185
0
votes
0
answers
36
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Integration measure for a strange substitution
I have a 2D integral over a momentum vector, i.e. $\int dp_x dp_y$ and the substitution for this is given by
$$ \xi = |\vec{p}| + |\vec{p} + \vec{q}| , \, \, \, \eta = |\vec{p}| - |\vec{p} + \vec{q}|$$...
- 1