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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.

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. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Read more on wikipedia's entry Surface integral.

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Tricky surface integral of vector field

The following problem comes from a vector calculus exam. Let $$ S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\} $$ be an embedded surface with the orientation corresponding to the positive $\bf{OZ}$ direction, and let…
Sam Skywalker
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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 Stokes' Theorem, it seems to me that we evaluate…
hs_math_enjoyer
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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 asked, although we integrate over the unit sphere, the…
user505183
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What is $\int_0^1\int_0^1\frac{x^a(1-y)^b}{(1-xy)^c}\,{\rm d}x\,{\rm d}y$?

$\quad$What is the value of the integral $$\int_0^1\int_0^1\frac{x^a(1-y)^b}{(1-xy)^c}\,{\rm d}x\,{\rm d}y?$$ for nonnegative integers $a,b,c$? $\quad$For example, setting $b=4$, $c=3$, and letting $a$ vary, it seems that…
Akiva Weinberger
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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 of the curve we are integrating over. This can…
14tim4
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In $\mathbb{R}^3$, does $dS = \sqrt{dx^2dy^2+dy^2dz^2+dx^2dz^2}$ hold in surface integrals?

I'm wondering whether the formula above is true or not. It looks like an extension of $dl = \sqrt{dx^2+dy^2+dz^2}$ used in line integrals. I don't see it in my textbook, so it is likely wrong since the formula looks too nice to omit. But if it is…
Anh Quang Bùi
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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 using cylindircal polar…
Censacrof
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Is there a tangential surface integral?

In $\mathbb{R}^2$, we have two different types of line integrals, the tangential line integral $$\int_C \mathbf{F}\cdot d\mathbf r$$ and the normal line integral $$\int_C \mathbf{F}\cdot \mathbf n \,ds.$$ To give a motivation, these two different…
Xiao
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Evaluating $\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$

How can I evaluate this integral? $$\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$$ I know from Wolfram Alpha that the answer is $\frac13\ln2+\frac16\approx0.39772$. However, doing this explicitly seems like a nightmare. Is…
Akiva Weinberger
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Why does the magnitude of the cross-product of a and b give the area of a parallelogram spanned by a and b?

I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working with surface integrals where the divergence…
User001
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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 parametrization that came to my mind was:…
user128422
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Use the divergent theorem to verify the volume of a circular cone

Let $T$ be a region with boundary surface $S$ such that $T$ has volume $$V=\frac{1}{3} \iint_S (x dydz +ydzdx+zdxdy)$$ use this equation to verify the volume of a circular cone with height $h$ and radius of base $a$ is $V = \pi a^2…
Diane Vanderwaif
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How can I determine if a vector is pointing inwards or outwards a surface?

Let $\gamma:\left[0, \frac\pi2\right] \to \mathbb{R^2}$ be a curve defined by $\gamma(\theta) = (\cos^2(\theta) \sin(\theta) , \cos(\theta) \sin^2(\theta))$. The following is the image of $\gamma$: where I have labeled…
Davide Masi
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( Proof Explanation ) Show that a certain system preserves the weighted area $ (dx \wedge dy)/xy$

I already told few questions ago that I'm currently reading an abstract about the Lotka Volterra differential equations. But now I have a proof, where I need explanations. Consider: $$ \dot{x} = -xy\frac{\delta H}{ \delta y} , x(0) = \hat{x} $$ $$ …
RukiaKuchiki
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When is a surface integral equal to double integral over projection? A verification of Stokes' Theorem. Intuition and relation to Green's Theorem.

I'm helping a calculus student. It's been awhile since I've done some vector calculus. I know Stokes' Theorem in terms of differential forms and manifolds with boundary, but I've forgotten much of Stokes' Theorem in the undergraduate setting. The…
user636310
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