All Questions
Tagged with integration multivariable-calculus
5,459
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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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Integration by parts on an area
I'm reading an Engineering book. All I can think of is integration by parts
$$\int_{\Omega} \dfrac{\partial M_y}{\partial x}v\text{d}\Omega = M_y v|_{?}^{?} - \int_{\Omega} M_y\dfrac{\partial v}{\...
- 175
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Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function
I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
- 5,064
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Integral calculated directly and with Gauss Green formula
I checked on Wolfram Alpha that the directly calculated integral was correct, when I go to use the Gauss Green formula, I get $0$ as a result, when instead, I ...
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Theorem 16.5, Munkres' Analysis on Manifolds [closed]
In Munkres' Analysis on Manifolds, page 142 Theorem 16.5 it states:
$$\int_{D}f\leq\int_{A}f$$
at the end of that page.
Here, $D=S_{1}\cup\cdots\cup S_{N}$ is compact since $S_{i}=Support\phi_{i}$ and ...
- 174
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Integral in polar coordinates
I was wondering if I had set up this integral correctly. If anyone could help me, I would greatly appreciate it. I am available in case there are any unclear things!
$$\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}...
- 3,040
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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 ...
- 3,040
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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 $$ f(d_1) = \frac{\int_0^\pi \int_0^\pi \int_0^{2\pi} 4\pi sin^2(\frac{\theta}{2})(1+(1-cos\theta )(v_1^2-1)) e^{k(d_1 + d_2 + d_3 + (1-...
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between the plane $z=0$ and the cone $x^2+y^2=z^2.$
I tried solving this problem as follows:
Equation of the cylinder $x^2+(y-a)^2=...
- 3,040
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
Find the volume of the solid inside the cylinder $x^2+y^2-2ay=0$ and between $z=0$ and the paraboloid $4az=x^2+y^2$ equals $\frac{3\pi a^3}{8}.$
I tried solving this problem as follows:
The equation ...
- 3,040
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$.
Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
- 5,468
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What does the integral $\int_0^V d\vec{x}$ mean/represent and how is it calculated?
My question is what does an integral such as
$$\int_0^V d\vec{x}_1$$
mean exactly?
Here is the context in which this type of integral arose.
I am following a thermodynamics course that has a section ...
- 5,023
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integration by parts $\int_{\mathbb{R}^N} -\Delta u (u-M)\varphi dx$
I'm trying to understand how it integrates by parts the following integral. This belong to the academic paper Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$ with DOI 10.4171/JEMS/217. ...
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Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
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