Skip to main content

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.

405 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
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 ...
user avatar
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
  • 71
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
0 answers
65 views

Solution verification of flux integral $\iint_DF\cdot\hat{n}dS$ with $F=(2x,y,z)$

Compute the flux of $F=(2x,y,z)$ through the surface $$ r = u^2v\,\hat{\imath} + uv^2\,\hat{\jmath} + v^3\,\hat{k}, \quad 0\leq u \leq 1, \quad 0 \leq v \leq 1 $$ My approach: is it correct?
Math Lover's user avatar
4 votes
0 answers
110 views

Asymptotic expansion of a double integral with a Bessel function

I am attempting to figure out the first term of the asymptotic expansion at $t \to \infty$ of the following double integral: $$ I(\tau) = \iint_{[\sqrt{\tau},\tau]^2} \frac{e^{2(u+v)} I_1(2 (2\...
user2673062's user avatar
4 votes
0 answers
71 views

Fitting an Ellipse to an Ink Drop on The Cloth With Constraints

I am looking to fit an ellipse on to some 2d spacial density function. So for an analogy if we put ink onto some cloth the ink dye will spread maybe in the shape of circle and we can simply solve the ...
GENIVI-LEARNER's user avatar
4 votes
0 answers
400 views

Surface integral enclosing all of space? Improper surface integral convergence?

Consider a surface integral over a closed surface $S$ in $\mathbb{R}^3$ $$ \iint_S V\vec{E} \cdot d\vec{S}$$ (I'm writing out $V$ and $\vec{E}$ to make connection with physics textbooks, and to show ...
DWade64's user avatar
  • 1,318
4 votes
0 answers
134 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 ...
Deepabali Roy's user avatar
4 votes
1 answer
571 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}...
bkocsis's user avatar
  • 1,258
3 votes
0 answers
67 views

Struggling with a Calculus III problem :$\iint x+y\ dS,$ where $S(u,v)=2\cos(u)\vec i+2\sin(u)\vec j+v \vec k$ and $0\le u\le \pi/2;\;0 \le v \le9.$

So I have had Calc III many moons ago but I cannot seem to solve this problem for my son who is taking it now. Worked it four ways and got four different answers. Hoping someone here can set me ...
Transonic's user avatar
3 votes
0 answers
117 views

Verifying Gauss divergence theorem

The question says that: If $\vec{F}=(2x^2-3z)\vec{i}-2xy\vec{j}-4x\vec{k}$, we are supposed to calculate $\iiint \operatorname{div} \vec{F}\, dV$, where $V$ is the closed region bounded by the planes $...
S.S's user avatar
  • 1,229
3 votes
0 answers
32 views

Surface Integral - Better Parametrization?

So I'm supposed to calculate $$ \iint_S (x^2+y^2)dS $$ Where $S$ Is the part of the plane $z=2x+2y-1$ which is inside the paraboloid $z=x^2 +y^2$. The way I proceeded was to parametrize $$ S: \begin{...
H44S's user avatar
  • 856
3 votes
0 answers
155 views

Evaluating the Surface Integral on a Sphere for a scalar function (integral is involved in PDE)

I will begin by saying that I don't want to dissuade anyone who doesn't know PDE from helping, so if you're just here for the integral, you can skip down to "Where I'm Stuck" (For future reference: $\...
ninja cat's user avatar
3 votes
0 answers
123 views

Surface integral in a non-centered circle.

I am working in a problem of fluxes of vector maps, and after having applied the divergence theorem I have obtained the following integral: $\Phi = \int_{x=-R}^{x=R} \int_{y=-\sqrt{R^2-x^2}}^{y=\...
alexmolas's user avatar
  • 169
3 votes
0 answers
215 views

Evaluating the surface integral over subset of Cylinder surface

I am preparing for the GRE math subject test. And I am not sure whether my solution to this surface integral problem is correct. Problem Image. Surface $S$ is part of $x^2 +y^2 = 1$ between ...
Jason's user avatar
  • 254
3 votes
0 answers
52 views

Evaluating $\iint_S z \, dS $

My professor gave us a handout for these problems and I'm not sure if I'm doing these correctly. Could someone tell me if I'm on the right track? Problem $\iint_S z \,\mathrm{d}S$ $z=x^2+y^2$ and $...
z400jt's user avatar
  • 127
3 votes
0 answers
49 views

Computing the average of $\prod_i (1-\frac{|x_i|}{L})$ on the surface of the unit $n$-sphere

I'm trying to compute the expected value of $ \prod_i (1-\frac{|x_i|}{L})$ on the surface of a $n$-dimensional sphere. A first step could be to integrate only on the first quadrant to take out the ...
etal's user avatar
  • 63
3 votes
0 answers
62 views

Apparent (minor) error in Cauchy's article on pressure or tension in a solid body

In his article De la pression ou tension dans un corps solide [On the pressure or tension in a solid body], Cauchy introduces a theory that allows to define Cauchy stress tensor. It looks like he ...
Alexey's user avatar
  • 2,162
3 votes
0 answers
156 views

What happens to tangential gradient when flattening a surface

The tangential gradient $\nabla_\tau f$ associated to a surface $S$ is defined as the projection of a suitable extension $\nabla f$ to the tangent plane to that surface. It seems reasonable to think ...
analyst's user avatar
  • 31
2 votes
0 answers
176 views

Surface integral where $S$ is the surface of a solid bounded by cylinder $x^2+z^2=4$ and planes $y=0$ and $y=3$.

$$\iint_{S}\langle-3z^2,1-x,(y-2z)\rangle\,d\vec{s}$$ I am trying to solve this surface integral where $S$ is the surface of a solid bounded by cylinder $x^2+z^2=4$ and planes $y=0$ and $y=3$, I ...
12 3's user avatar
  • 47
2 votes
0 answers
48 views

Surface integral of a complex Log function

I am trying to calculate the surface integral of a complex Log function i.e. $$ \int\int_{|z|<1}{Log(x+i y-(x_0+iy_0)))dxdy}$$ where $z=x+iy$ and $x_0,y_0 \in \mathbb{R}$ . I know that for analytic ...
O.s.'s user avatar
  • 21
2 votes
0 answers
174 views

Can I calculate the area of a closed surface on a sphere by its boundary using Stokes' Theorem?

So I (just learned that I) can get the area of any closed area $S$ with boundary $\partial S$ on the flat plane through Stokes' (or in the simplified case Green's) theorem by using a vector field with ...
Fl0wless's user avatar
  • 121
2 votes
0 answers
52 views

Volume of a square pyramid with a curved base

Consider a square pyramid of base length $a$ and height $H$ with vertices at coordinates $\left(\pm \frac{a}2, \pm \frac{a}2\right)$ and $(0,0,H)$. Assume that $f : \Bbb{R}^2 \to \langle 0,H\rangle$ ...
mechanodroid's user avatar
  • 46.6k
2 votes
0 answers
98 views

Surface Integral of Vector Field $\textbf{A}=(xz,0,yz)$

I have to solve the following problem: A cylinder of radius R in $\mathbb{R}^3$ with axis along the $z$-axis of a Cartesian coordinate system $(x,y,z)$ can be parametrised as $\textbf{x}(\theta,z) = (...
Toniiiic's user avatar
  • 205
2 votes
0 answers
37 views

Derivation of the surface element $dA$ on a parameterized surface?

Let $S\subseteq\mathbb R^3$ be some smooth surface (smooth 2-submanifold of $\mathbb R^3$) with global parameterization $\psi\colon\mathbb R^2\supseteq U\to S,(s,t)\mapsto\psi(s,t)$. The surface area $...
Cubi73's user avatar
  • 775
2 votes
0 answers
64 views

Flux of the horizontal Electric field through a hemisphere

Suppose I've a hemisphere and an electric field passing horizontally through this hemisphere. I need to find the flux of this field through this hemisphere. I can easily consider the electric field to ...
Nakshatra Gangopadhay's user avatar
2 votes
0 answers
133 views

Line and Surface Integral with the Dot Product replaced with a Cross Product

Having recently studied magnetostatics, I came across the Biot-Savart law, which is based on the line integal over a current distribution in a curve $C$: $$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\...
Angelo Di Bella's user avatar
2 votes
0 answers
110 views

Help visualizing the cross product when calculating surface area of a sphere with cylindrical coordinates

I don't have enough reputation to comment on the answer to the question asked here so I had to make a new question to ask a follow-up. I know that the cross product gives an area but I'm having ...
vwlau's user avatar
  • 21
2 votes
0 answers
38 views

Trouble calculating/understanding this surface integral

Suppose $f\in C^3(\partial U)$ where $U \subset \mathbb R^3$ is an open, bounded and connected set with regular boundary. For some $s>0$ and $\epsilon>0$ sufficiently small, we define $A_{\pm}:=\...
kaithkolesidou's user avatar
2 votes
0 answers
53 views

Substitution rule for the surface measure on a $C^1$-submanifold

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $\Omega$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$. Assume, for simplicity, that $\Omega$ is described by a single chart, i.e. ...
0xbadf00d's user avatar
  • 13.6k
2 votes
0 answers
101 views

Computing a flux through only a portion of a surface

Let $V=\{(x,y,z)\in\mathbb{R}^3: 1\leq x^2+y^2+z^2 \leq 2, \ z^2 \geq x^2+y^2 \}.$ Now, I would picture this $V$ as being the intersection between a cone and the surface $S$ given by a 3D ball of ...
ggeolier's user avatar
  • 579
2 votes
0 answers
66 views

Generalized gaussian integral on a sphere

I am interested in the calculation of the following integral: \begin{equation} I=\int_{\sum_{i=1}^Nx_i^2=R^2} \mathrm d x \;e^{-\frac 12 x^T A x} \end{equation} where $A\in \mathbb M_N(\mathbb R)$ is ...
Graz's user avatar
  • 683
2 votes
0 answers
105 views

Evaluating surface integral where vector field is undefined

Let $$\vec{D} = \frac{K_1}{r}\hat{\phi} - \frac{K_2}{r\sin \theta}\hat{\theta}$$ where $K_1$ and $K_2$ are constants. Find $I =\oint \vec{D}.d\vec{S}$ where $S$ is a closed surface shown below: My ...
S.H.W's user avatar
  • 4,399
2 votes
0 answers
44 views

To evaluate a surface integral

I was stuck in a question as follows: $$\int_0^R\int_0^R\frac{(R^2-x^2)(R^2-y^2)}{(x+y)^2}dxdy$$ I tried a lot to simply this but I was unable to do so. I am encountering surface integrals for the ...
shsh23's user avatar
  • 1,135
2 votes
0 answers
143 views

How to define a 'surface measure' on a smooth boundary

Let $U$ be a bounded open set in $\mathbb{R}^n$ with a $C^1$ boundary $\partial U$. Then I encounter expressions like $L^p(\partial U)$. I know how to define the surface measure for $S^{n-1}$; but ...
Keith's user avatar
  • 7,715
2 votes
0 answers
125 views

How do you find the surface area of a Steinmetz solid?

I have read a lot about this on MSE before, but all simply briefly mention the possibility of doing it through line integrals and double integrals. But none of have actually explained in detail how it ...
Eliot Behr's user avatar
2 votes
0 answers
53 views

Determine the area of a semisphere cut by a cylinder

Semisphere has the ecuation $z=\sqrt{16-x^2-y^2}$ and the cylinder $x^2+(y-2)^2=4$ Should I parametrize the sphere with trig functions or leave it as $\vecΣ = (x,y,\sqrt{16-x^2-y^2})$? How do I ...
Juani Elias's user avatar
2 votes
1 answer
67 views

Stokes Theorem aplication

uses the Stokes theorem to calculate the surface integral, $I=\int_{S}CurlA.ds$, where $A(x,y,z)=(2y,3x,-z^{2})$, for all $x,y,z\in \mathbb{R}$ and $S=\left\{(x,y,z) \in \mathbb{R^{3}}: x^{2}+y^{2}+z^...
user avatar
2 votes
0 answers
231 views

How to evaluate this integral over a slice of the unit disk?

I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$) $$ f(r) = a + \dfrac{b}{(1-r)^n}$$ I would like to integrate ...
Sanchises's user avatar
  • 546
2 votes
0 answers
622 views

Proof of cross product integral

I am working through Susan Lea's Mathematics for Physicists however I am stuck on problem 31)b): 31) Prove b) $\int_{S}(\hat{n}\times\vec{\nabla})\times\vec{u}\hspace{0.5mm}dA = \unicode{x222E}_{C}...
QuantumPanda's user avatar
2 votes
0 answers
90 views

My Generalized Mean Value Theorem

I'm trying to prove the following without the divergence theorem(for soon to be obvious reasons): Let $F: \mathbb{R^3} \to \mathbb{R^3}$ be a differentiable function. In $\mathbb{R}^3$, let $V$ be a ...
user avatar
2 votes
0 answers
42 views

Flux through surface (correct or not?)

I need to calculate the flux of the vector field $$\mathbf{v}=(x-y,\; x+2y,\; z)$$ through the surface $$S\, : \, \mathbf{r}(u, v)=(\frac{u^2}{\sqrt{2}},\, uv,\, \frac{v^2}{\sqrt{2}}),\;\; 0≤u≤1,\...
Filip's user avatar
  • 495
2 votes
0 answers
417 views

Surface Area in $R^n$

Let's say we have a closed domain $D \subset R^m$. We then define a function over $D$ explicitly $f:D\rightarrow \mathbb{R}^n$. We can also state that is our function $f$ is differentiable $k$ times....
Francois Wassert's user avatar
2 votes
0 answers
86 views

$\iint_{S} F.\eta dA$ where $F = [3x^2 , y^2 , 0]$ and $S : r(u,v) = [u,v,2u+3v]$

I had to evaluate $\iint_{S} F.\eta dA$ where $F = [3x^2 , y^2 , 0]$ and $S : r(u,v) = [u,v,2u+3v]$ where $0 \leq u \leq 2 , -1 \leq v \leq 1$. I solved like this - $\iint_{S} F(r(u,v)) . (r_{u} \...
BAYMAX's user avatar
  • 4,992
2 votes
0 answers
216 views

Surface Integral and why "r" from (polar) change of variables is missing from the integral?

I'm ask to find the surface area of the cylinder $x^2+y^2=2x$ limited by the cone $z=\sqrt{(x^2+y^2)}$ and the plane $z=0$ and . I know that the cylinder's center is at $(1,0)$ $x^2+y^2=2x$ ...
user avatar
2 votes
1 answer
110 views

Confusion with computing surface integral

Let $F(x,y,z) = \langle x, y, z^2 \rangle$, and $S$ be the unit sphere with radius $1$ about the origin. The question is to find the surface integral $\iint_S F \cdot \text{d} S$. Using the divergence ...
user285292's user avatar
2 votes
0 answers
72 views

Finding the heat flow across the curved surface of a cylinder #2

This question is the second part to a previous question on the same problem. I have the following problem: The temperature at a point in a cylinder of radius $a$ and height $h$, and made of ...
The Pointer's user avatar
  • 4,262
2 votes
0 answers
133 views

Flux of vector field $F(x,y,z)=(x,y,xe^{y+z^2})$ through conical surface

Calculate the flux of the vector field $F(x,y,z)=(x,y,xe^{y+z^2})$ through the surface of the cone $z^2=x^2+y^2$ between $z=0$ and $z=1$. The flux is just the respective surface integral. I think ...
user avatar
2 votes
0 answers
116 views

Notation for surface integrals over vector fields

Why do every question I see for a surface integral over a vector field $F$ under the surface $S$ be denoted by $$\int_{S} F \cdot n dS$$? As in, why does it only have a single integral? Where as once ...
Morris C.'s user avatar
  • 243
2 votes
1 answer
1k views

Determining the flux of a vector field across a surface

The problem reads: Compute $\int_S \, \vec{F}(x, y, z) \cdot \vec{n} \ dS$, where $\vec{F}(x, y, z) = \left\langle x\ln(xz), 5z, \frac{1}{y^2+1} \right\rangle$, $S$ is the region of the plane $12x-...
user369262's user avatar

1
2 3 4 5
9