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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.

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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 ...
Sam Skywalker's user avatar
18 votes
3 answers
533 views

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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17 votes
2 answers
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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 ...
OneGapLater's user avatar
13 votes
2 answers
2k views

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 ...
14tim4's user avatar
  • 411
11 votes
2 answers
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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 ...
Censacrof's user avatar
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11 votes
4 answers
921 views

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 \...
Xiao's user avatar
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10 votes
1 answer
471 views

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 ...
user128422's user avatar
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9 votes
4 answers
31k views

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 ...
User001's user avatar
9 votes
1 answer
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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 ...
Diane Vanderwaif's user avatar
8 votes
1 answer
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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\...
RukiaKuchiki's user avatar
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7 votes
4 answers
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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 ...
Resquiens's user avatar
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7 votes
1 answer
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Which is the correct way to compute this surface integral?

I am trying to find a surface integral $$\iint_Syz\ dS$$ of a cylinder segment where $S$ is the portion of $x^2 + y^2 = 1$ with $x ≥ 0$ and $z$ between $z = 2$ and $z = 5 − y$. I thought that there ...
dlp's user avatar
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1 answer
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Applying surface integral on the Mobius strip

I'm trying to apply the surface integral on the Mobius strip. I know the Mobius strip's surface area can be easily calculated by getting the area of a piece of paper before it got twisted but this is ...
7_G.S.N's user avatar
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7 votes
0 answers
318 views

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 ...
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6 votes
1 answer
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Integral over a surface in 4-dimensions

Consider the integral of a function $f(x,y,z)$ over a surface embedded in 3 dimensions. The surface has a parameterization: $$g(u,v) = (x(u,v), y(u,v), z(u,v)) $$ The integral is given by: $$ \iint_{...
DWade64's user avatar
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6 votes
3 answers
2k views

Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find $$\color{red}{\iiint_\limits{\large\text{volume}\,\tau}\nabla\cdot\...
BLAZE's user avatar
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6 votes
1 answer
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Detail of a proof "Sobolev inequality $\Rightarrow$ Isoperimetric inequality".

From: Sobolev inequality: For all $u\in C_c^{\infty}(\mathbb{R}^n)$ $$\|u\|_{L^{\frac{n}{n-1}}(\mathbb{R}^n)}\leq C \|\nabla u\|_{L^1(\mathbb{R}^n)}.$$ I want to prove: Isoperimetric inequality:...
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6 votes
1 answer
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Divergence Theorem when Surface isn't closed

So we essentially want to evaluate $$\iint_S \vec{F} \cdot d\vec{S},$$ where $\vec{F} = \langle 2x+y, x^2+y, 3z \rangle$ and $S$ is the cylinder $x^2+y^2=4$, between the surfaces $z=0$ and $z=5$. We ...
user avatar
6 votes
1 answer
108 views

Purpose of the "$\vec{F} \cdot \text{d}\vec{S}$" notation in vector field surface integrals

When describing surface integrals in vector fields, it's common to use the notation $$\iint_S \vec{F} \cdot \text{d} \vec{S}$$ as a shorthand for $$\iint_S \vec{F} \cdot \vec{n}\, \text{d}S$$ This ...
Mu Prime's user avatar
  • 124
6 votes
2 answers
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Calculating Surface Area using Differential Forms

I'm trying to reconcile the definition of surface area defined using manifolds vs the classic formula in $\mathbb{R^3}$, but it seems like I'm off by a square. In Spivak's Calculus on Manifolds, the ...
Snowball's user avatar
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6 votes
2 answers
1k views

Extension of Poincaré inequality: $\|u\|_{L^2(\Omega)}\leq C\|\nabla u\|_{L^2(\Omega)}$ when $u$ vanishes in $\Gamma\subseteq\partial\Omega$.

Recall Poincaré inequality: Let $\Omega$ be a bounded open set in $\mathbb{R}^n$. Then there is a $C=C(\Omega,n)>0$ such that $\|u\|_{L^2(\Omega)}\leq C\|\nabla u\|_{L^2(\Omega)}$ for all $u\in ...
user39756's user avatar
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6 votes
0 answers
95 views

Integrating product of harmonic functions over sphere

Im a math major student and taking a course in multivariable calculus. I straggled with the following homework exercise. Let $u,v :\mathbb{R}^n \to \mathbb{R}$ be harmonic functions (i.e. $\Delta u,\...
Haruki's user avatar
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6 votes
1 answer
147 views

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 ...
constant94's user avatar
5 votes
2 answers
5k views

Surface area of sphere within a cylinder

I have to Compute the surface area of that portion of the sphere $x^2+y^2+z^2=a^2$ lying within the cylinder $\Bbb{T}:=\ \ x^2+y^2=by.$ My work: I start with only the $\Bbb{S}:=\ \ z=\sqrt{a^2-x^...
Qwerty's user avatar
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5 votes
6 answers
2k views

Surface integrals in spherical coordinates

If I am given a surface in spherical coordinates $(r,\theta,\varphi)$, such that it is parametrised as: $$ \begin{align} r&=r(\theta,\varphi)\\ \theta&=\theta\\ \varphi&=\varphi \end{align}...
atapaka's user avatar
  • 507
5 votes
1 answer
4k views

Conversion of Surface integral to a suitable Volume integral.

While deriving the Euler's equations of motion in case of Fluid dynamics, I came across this part - Here $p$ denotes the hydrostatic pressure(scalar function) I am unable to understand how it ...
BAYMAX's user avatar
  • 4,992
5 votes
3 answers
2k views

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 ...
Bruno Marcelo Goicochea Vilela's user avatar
5 votes
2 answers
4k views

Calculating surface area of a sphere with cylindrical coordinates

Calculating the surface area of a sphere of radius $a$ with spherical coordinates is quite easy ($4\pi a^2$). I'm trying to do the same with cylindrical coordinates, $\rho$, $\theta$, $z$, (just for ...
user297517's user avatar
5 votes
2 answers
826 views

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 ...
Lozansky's user avatar
  • 1,035
5 votes
1 answer
837 views

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 ...
coderodde's user avatar
  • 291
5 votes
1 answer
163 views

Scalar integrals in higher dimensions

The thing I want to do The typical vector calculus course defines: A bunch of integrals of vector fields in $\mathbb R^2$ and $\mathbb R^3$: line integrals of a vector field along a curve, flux ...
Misha Lavrov's user avatar
5 votes
2 answers
124 views

If $f$ is Lebesgue integrable on an open set $U$ is it integrable over the surface of a submanifold contained in $U$?

Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open and $M\subseteq U$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ Let $f\in\mathcal L^1(U)$ and $\sigma_M$ denote the surface ...
0xbadf00d's user avatar
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5 votes
1 answer
2k views

Find the flux out of a tetrahedron.

The problem: Find the flux $\textbf{F} = 3x\hat{i} + z\hat{j}$ out of the tetrahedron closed in by the plane $5x + 3y + 3z = 4$ and the xy, xz and yz planes. My (wrong) solution: I calculated the ...
Viktor's user avatar
  • 139
5 votes
1 answer
100 views

How to evaluate double integrals of a surface over a specific region?

I found this exercise while exercising for the exam: Let $T$ $\subset$ $R^2$ be the triangle with these vertices $(0,0), (2,0), (0,1)$ and let $\Omega$ be the surface defined like this: $\Omega$ = {$(...
Tindaro Mirabile's user avatar
5 votes
1 answer
786 views

Generalizing a surface integral to 4 dimensions

I am trying to evaluate a surface integral, but instead of using a surface in $\mathbb{R}^3$, using a surface in $\mathbb{R}^4$. That is to say, $\oint_S f(x,y,z,w)\,dS$, where S is given by some $r(u,...
user avatar
5 votes
2 answers
125 views

Area of a surface using integration. Confusion with aspect of formal definition.

My textbook explains that, when finding the area of a surface using integration, we approximate each surface element by $$\left| \Delta u \dfrac{\partial \overrightarrow{r}}{\partial u} \times \Delta ...
The Pointer's user avatar
  • 4,262
5 votes
1 answer
2k views

steps to calculate the space surface area cut by a cylinder(see the picture)

The space surface(in yellow) $ x^2+y^2 = 2az\ $ is cut by a cylinder(in green) $x^2+y^2=3a^2 (a>0)$ How to calculate the cut out part area $A$? I think the part is between the two planes $z=0$ and $...
canoe's user avatar
  • 411
5 votes
2 answers
431 views

Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
user avatar
5 votes
1 answer
890 views

Integral of divergence over a closed surface

I am reading a paper, where an integral of a divergence over a closed surface is used without proof. $\oint_S [\nabla \cdot \vec{v}(\vec{r})] d\vec{s} = 0$, where $\vec{v}$ is tangential to the ...
Aleksejs Fomins's user avatar
4 votes
1 answer
2k views

Surface integral of normal components summations on a sphere

Whilst studying a book on fluid dynamics I came across a curious footnote comment which is essential in another derivation. The footnote states the following two identities: $$ \frac{1}{|S|}\oint_S\! ...
Robert Manson-Sawko's user avatar
4 votes
2 answers
2k views

Surface integral over a sphere of inverse of distance

Let $S$ be a sphere in $\mathbb{R}^3$ of radius $r$ centered at the origin and $x_0\not\in S$. Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x)=\Vert x-x_0\Vert$. I'm asked to compute the (surface) ...
Brandon's user avatar
  • 3,185
4 votes
1 answer
3k views

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-...
User001's user avatar
4 votes
2 answers
2k views

Surface area of the ellipsoid $\frac{x^2}{16}+\frac{y^2}{8}+z^2=1$

My professor gave us this question on a calculus II quiz. One of my calculus III pals suggested I use surface integrals, but that tool is not available to us (I don't know how to use it yet, nor do my ...
Carly Sawatzki's user avatar
4 votes
5 answers
137 views

Computing $\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y$

Computing $$I=\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y,$$ where $S$ is the upper side of the plane $x+y+z=1$ located inside the interior of the ...
MKCCT's user avatar
  • 79
4 votes
2 answers
132 views

How to interpret a double integral?

What can be the geometrical meaning of $$ \iint_R dA ~~~~~~~~~~ (1)$$ ? It is the particular case of $$\iint_R {f(x,y)} dA$$ where $f(x,y) = 1 $. What I have found from my search is that (1) ...
Knight wants Loong back's user avatar
4 votes
3 answers
943 views

Relationship between $(n - 1)$ forms and flux of a vector field across a hypersurface

I am currently studying about differential forms and want to deduce the Divergence Theorem (the one in $\mathbb{R}^n$) from the general Stokes' Theorem, which is obtained by taking $$\omega = \sum_{i=...
Theorem's user avatar
  • 2,948
4 votes
1 answer
82 views

Show that $\int_{S^{n-1}_+}\frac{1}{\left \langle y,a \right \rangle^n}dS(y)=\frac{1}{(n-1)!}\frac{1}{a_1\dots a_n}$

Let $\mathbb{R}^n_{+}$ be the set of points in $\mathbb{R}^n$ with all coordinates positive. Let $S^{n-1}_{+}=S^{n-1}\cap \mathbb{R}^n_{+}$. Let $a=(a_1, \dots ,a_n)\in \mathbb{R}^n_{+}$. Show that: ...
35T41's user avatar
  • 3,387
4 votes
1 answer
4k views

How do you determine the boundary curve for Stokes' Theorem

In simple examples, such as with a paraboloid, determining the boundary curve is simple enough. However, when I am faced with more complex examples, I seem to get lost and do not know the proper way ...
Rob's user avatar
  • 511
4 votes
1 answer
4k views

What is the physical meaning behind the surface integral

For example, I know that the physical meaning behind a standard, single integral is the area under the curve (with respect to the x or y axes). Likewise, the a line integral can be physically ...
God's Drunkest Driver's user avatar
4 votes
2 answers
357 views

Curl of a vector field.

Let S be a piecewise smooth oriented surface in $\mathbb{R}^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
Rafa Fafa's user avatar
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