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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362 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 ...
11
votes
2answers
666 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 ...
10
votes
2answers
914 views

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 ...
9
votes
1answer
235 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 ...
8
votes
2answers
7k views

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 ...
7
votes
4answers
1k views

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 ...
7
votes
3answers
334 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 \...
7
votes
1answer
64 views

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 ...
6
votes
3answers
315 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\...
6
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1answer
420 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 ...
6
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1answer
99 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 ...
5
votes
2answers
187 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 ...
5
votes
3answers
742 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 ...
5
votes
2answers
418 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 ...
5
votes
1answer
913 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 ...
5
votes
1answer
649 views

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 ...
5
votes
2answers
80 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 ...
5
votes
1answer
448 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 $...
5
votes
1answer
217 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)}{\...
4
votes
2answers
702 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 ...
4
votes
1answer
2k 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 ...
4
votes
2answers
169 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 ...
4
votes
1answer
536 views

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:...
4
votes
2answers
142 views

Computing line integral using Stokes´theorem

Use Stokes´ theorem to show that $$\int_C ydx+zdy+xdz=\pi a^2\sqrt{3}$$ where $C$ is the curve of intersection of the sphere $x^2+y^2+z^2=a^2$ and the plane $x+y+z=0$ My attempt: By Stokes´ theorem ...
4
votes
1answer
160 views

Two properties of surface integrals.

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^1$ boundary. I know the following (possible) definition for $$\int_{\partial\Omega}u\,d\sigma.$$ For each $x_0\in\partial\Omega$, there ...
4
votes
1answer
66 views

Is $C_c^{\infty}(\mathbb{R}^n)$ dense in $L^p(M,d\sigma)$, $1\leq p<\infty$, where $M$ is an $n-1$ regular surface in $\mathbb{R}^n$?

I know that, given an open set $\Omega\subseteq\mathbb{R}^n$, $C_c^{\infty}(\Omega)$ (smooth functions with compact support) is dense in $L^p(\Omega)$, $1\leq p<\infty$. Let $M$ be a smooth $n-1$ ...
4
votes
1answer
157 views

Use Stokes' theorem to show that the line integral has the value given.

Problem 9 from Apostol's Calculus 2 book, page 443: Use Stokes' theorem to show that the line integral has the value given: $$\int_C (y^2 + z^2)\,dx + (x^2 + z^2)\,dy + (x^2 + y^2)\,dz = 2\pi ...
4
votes
1answer
623 views

Surface area of sphere using Dirac delta

This question is related to this one. Suppose I want to calculate the surface area $S(R)$ of a sphere of radius $R$. I can express $S(R)$ as $$S(R)=\int_{\mathbb{R}^3} \delta (\| \vec x \|-R) \ d \...
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0answers
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+50

( 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\...
4
votes
2answers
50 views

Theorem of Pappus

Given a surface of revolution $S$ which can be parametrized by the map $$ \mathbf x(u,v) = (f(v)\cos u,f(v)\sin u,g(v)), $$ over the open set $U =\{(u,v) \in \mathbb R^2 \mid 0 < u < 2\pi, a <...
4
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0answers
112 views

Unable to evaluate this surface integral

Question is to find the surface integral $$\iint p(x^4 + y^4 + z^4) \, \mathbb dS.$$ The given surface is $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$$ and $p$ is the length of the ...
4
votes
1answer
225 views

Asymptotics of Gaussian integral over the unit sphere

I would like to evaluate the integral asymptotically over the unit sphere surface $$ Z =\int e^{a \cos^2 \theta + b \sin^2\theta\cos2\phi + c\cos\theta} d\Omega = \int\limits_{0}^{\pi}\int\limits_{0}...
3
votes
2answers
132 views

Show that $\iint_S (x^2+y^2) dA = 9 \pi /4$

In exam it was asked to show that $$\iint_S x^2+y^2 dA = 9 \pi /4$$ for $$S = {\{(x, y, z) | x>0, y>0,3>z>0, z^2 = 3(x^2 + y^2)}\}$$ I have tried many times but I don't get the $9 \pi ...
3
votes
4answers
218 views

Parametrizing the surface $z=\log(x^2+y^2)$

Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$. I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times \textbf{T}_v|...
3
votes
2answers
58 views

Surface integral - cone below plane

After several years I suddenly need to brush up on surface integrals. Looking through my old Calculus book I have been attempting to solve some problems, but the following problem has really made me ...
3
votes
1answer
223 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 ...
3
votes
2answers
478 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) ...
3
votes
2answers
1k 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^...
3
votes
1answer
52 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: ...
3
votes
1answer
2k views

Vector Normal to a Paraboloid

This might be a stupid question, but I think I am misunderstanding something. I was working on a calculus question which requires the calculation of a vector normal to a paraboloid. $z=a^2-x^2-y^2$ ...
3
votes
1answer
265 views

Counter clockwise flux integral

There's a function $\ \varphi :(0,+\infty) \to \Bbb R$ and another function $\ u:\Bbb R^{2} \to \Bbb R$ defined as $\ u(x,y):=\varphi(x^2+4y^2)$. For some $\ t>0$, let $\ E_{t} = \{{(x,y): x^2+4y^2=...
3
votes
3answers
309 views

What surface is represented by the following equation

$$\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}+\sqrt[3]{z^{2}}=1$$ Taking cubes of both sides only leads to a more complicated formula. How should one interpret this one. And, also if you could point me to a tool ...
3
votes
3answers
95 views

How to evaluate $\int_S(x^4+y^4+z^4) \, dS$ over surface of the unit sphere.

Question. Let $S$ denote the unit sphere in $\mathbb{R}^3$. Evaluate: $$\int_S (x^4+y^4+z^4) \, dS$$ My Solution. First I parametrize $S$ by $$r(u,v)=(\cos v \cos u, \cos v \sin u, \sin v)$$ $0\le u \...
3
votes
1answer
287 views

Evaluating surface integral (1) directly and (2) by applying Divergence Theorem give different resoluts

I have a vector field $\mathbf{A}(r) = \frac{1}{r^2}\mathbf{a}_r$. I am interested in finding the flux through a sphere enclosing some volume with radius $R$ and center at $r=0$. Calculating the ...
3
votes
1answer
2k 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 ...
3
votes
2answers
1k views

Parametrization of a tetrahedron?

I need to do a surface integral over the surface bounded by $x + 2y + z = 4$ and the coordinate planes. How do you go about parametrizing this surface? My first thought was with $x = x$, $y = y$, and $...
3
votes
2answers
108 views

Calculate the volume bounded by the surface $(x^2+y^2+z^2)^2 = x$

I need to solve: Calculate the volume bounded by the surface $$ (x^2+y^2+z^2)^2 = x $$ and not sure on how to do it. If I move to spherical coordinates, I get that the equation gives: $$ r^4=r\...
3
votes
1answer
1k 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-...
3
votes
1answer
69 views

Calculate surface integral

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 ...
3
votes
1answer
22 views

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 $$ \int_{S^{n-1}}dS=\frac{2\pi^{\frac{n}{2}}}{\Gamma\left(\frac{n}{2}\right)} $$ ...