Questions tagged [surface-integrals]

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.

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Investigate maxima of Gaussian integral over sphere.

Let $\alpha>0$ be a positive parameter and consider the function $$f(x) = \int_{\mathbb S^{n-1}} e^{-\alpha \left\lVert x-y \right\rVert^2} dS(y)$$ for $x \in \mathbb R^n.$ So, since this was ...
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Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent ...
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Faster way to calculate the area of this surface

I have to solve this exercise: Find the area of the portion of the surface $z=xy$ included between the two cylinders $x^2+y^2=1$ and $x^2+y^2=4$ what i did so far: I parameterized the surface ...
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Evaluating the Surface Integral $\iint_{x^3+y^3+z^3=a^3} \frac{\bf{x}}{||\bf{x}||} \cdot d\bf{S}$

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}) \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
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Difference between Stokes' Theorem and Divergence Theorem

Recently learnt the two and I really can't tell the difference. I'm not sure if I'm missing something, but it really seems to me that they evaluate the same thing just using different methods. With ...
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Why is the surface area of a sphere not given by this formula?

If we consider the equation of a circle: $$x^2+y^2=R^2$$ Then I propose that the volume of a sphere of radius $R$ is given by the twice the summation of the circumferences of the circles between the ...
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Calculating flux through a moving surface in a vector field that evolves with time

Suppose we are given a vector field $\mathbf{F} \colon \mathbb{R}^4 \to \mathbb{R}^3$ that evolves with time and describes the way, say, liquid particles move in a tank. Also, we are given a ...
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Where is my mistake? Calculating surface integral/Stoke's theorem

Let $F(x,y,z)= \begin{pmatrix} -y \\ 2x\\z \end{pmatrix}$ be a vector field and $A$ a hemisphere with $x^2+y^2+z^2=9$ , $z>0$ with a circular edge at the $x,y$- level with the unit normal ...
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Applying Stokes' theorem - what surface?

$\def\d{\mathrm{d}}$Determine the integral $$\oint_L \mathbf{A} \cdot \,\d\mathbf{r},$$ where $$\mathbf{A} = \mathbf{e}_x(x^2-a(y+z))+\mathbf{e}_y(y^2-az)+\mathbf{e}_z(z^2-a(x+y)),$$ and $L$ is ...
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Integrate a two form on the sphere

$$\int_S x\,dy\,dz+y\,dz\,dx+z\,dx\,dy,$$ where $S=\{(x,y,z)\in \mathbb{R}^{3}: x^2+y^2+z^2=1\}.$ Please, I don't know how to proceed. I will be thankful if you give me any hint, at first I thought ...
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Some questions about the normal vector and Jacobian factor in surface integrals,

I have some short questions on some lingering confusing concepts, specific to surface integrals: a) Is the surface integral in the Divergence and Stokes's Theorem the same thing? Both require a unit-...
I need some help with the following: Given $$f(x,y,z)=\left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac32}}, \frac{-z}{(x^2+y^2+z^2)^{\frac32}} \right),$$ calculate the ...
Integral over a surface area $\int_{S^{n-1}}x_{1}^{2}dS$
I want to evaluate the following integral: $$\int_{S^{n-1}}x_{1}^{2}dS$$ And I think i'm supposed to use the fact that $$\int_{S^{n-1}}dS=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)}$$ ...