All Questions
Tagged with integration multivariable-calculus
5,463
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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
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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.
$$\int_0^1\int_0^1\frac{\partial^2}{\partial\rho^2}\{s\rho^2C(\textbf{r}_2,ss^\prime\rho)...
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Looking for citation of a definition of multivariable integral
I need a definition of Multivariable Riemann Integral to cite in my article.
I've been searching different cited sources, for example Rudin, Folland, Stewart and Larson books.
In the first two I ...
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Am I correctly applying repeated integration by parts?
Say we have two compactly supported functions $f,g:\mathbb{R}^n\to\mathbb{R}$. I found myself computing
\begin{equation}
\begin{split}
\int_{\mathbb{R}^n}f\frac{\partial^{r}g}{\partial x^{\alpha_1}\...
- 5,166
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Need help with the steps and limits in this multivariable integration of a joint probability density function
I am not sure how to proceed with this double integration. I know this can be evaluated in a much easier way than solving the integral as its just the volume of a cube but I need help with the process ...
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Intergrating $ \int_{M} f (x.,y,z,w)\ d {\rm Vol}_3 $
I want to integrate $$ I = \int_{M} f (x.,y,z,w)\ d {\rm
Vol}_3 $$where $f(x,y,z,w) = (x+y)e^{z+w} $, and
$M = \{x+y+z+w = 1, x,y,z,w > 0\} $.
I need to find a parameterization of M; if I consider $...
- 391
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds
In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
- 174
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2
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66
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Integration by parts on an area
I'm reading an Engineering book. All I can think of is integration by parts
$$\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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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
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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
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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
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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!
$$\iint_D x^2+y^2dxdy, \quad D=\{1\...
- 179
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A volume problem in multivariable calculus that gives us $2$ different answers on $2$ different occasions.
Find the volume of the solid contained inside the cylinder $x^2+(y-a)^2=a^2$ and the sphere $x^2+y^2+z^2=4a^2.$
Now, I was able to solve this problem by evaluating $V=\int\int_D\int_0^{4a^2-x^2-y^2}...
- 3,065
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Evaluate $\iint_S\vec{F}$ where $S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above and below by two planes.
Use the divergence theorem to evaluate $$\iint_S\vec{F}$$ where $\vec F(x,y,z)=\langle xy^2,x^2y,y \rangle$ and $\vec S$ is the surface oriented outwards of the cylinder $x^2+y^2\leq 3$ bounded above ...
- 3,065
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How to analyze the behavior of this function?
I have a map from positive reals to positive reals, of the form $$ 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-...
