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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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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 ...
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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: $\...
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63 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=\...
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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 ...
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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 ...
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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 ...
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Surface area of a sphere above the plane $z=1$

The question is: Find the surface area of the part of the sphere $x^2+y^2+z^2=4$ that lies above the plane $z=1$. I got $4\pi(\sqrt3-1)$ but the answer key says $4\pi(\sqrt2-1)$. Am I doing something ...
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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}...
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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 ...
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computing fairly complicated integrals on the sphere

Assume $D$ is a diagonal matrix $2n\times 2n$. Let $J$ be the standard symplectic matrix, i.e.: $$J:=\begin{pmatrix} 0 &-\text{id}\\ \text{id} &0\end{pmatrix}.$$ I would like to compute the ...
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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,\...
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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....
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$\iint_{S} F.\eta dA$ 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} \...
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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 $(1,0)$ $x^2+y^2=2x$ ...
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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 ...
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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 ...
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662 views

derivative of a surface integral with moving boundary

Problem Let us assume surface $ \Sigma$ as a domain, with $\partial \Sigma$ as a moving boundary. $\Sigma$ is a portion of a sphere and $\partial\Sigma$ is a circle and its direction of movement is ...
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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 $...
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How to calculate a surface area of a river?

I am doing a math exploration and I was wondering if someone could help with this problem. What will I need to use in order to calculate SA of a river? What parts of math are used? What info will I ...
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Is it possible to calculate a surface integral of a vector field when the vector field is described in non-cartesian coordinates?

Every note and book I read about surface integrals of vector fields only show how to solve these integrals when the vector field is in Cartesian coordinates. I'm curious about what would be the right ...
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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. $\oint_S \nabla \cdot \vec{v}(\vec{r}) = 0$, where $\vec{v}$ is tangential to the surface ($\vec{...
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Value of an iterated integral

Show that $$\int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} \frac{u_{\phi \phi}}{\sin\theta} d\theta\,d\phi = 0 $$ Where $u$ is a function of $\theta$ and $\phi$. I am unable to show that this ...
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Surface area common to two perpendicularly intersecting cylinders

I need help to calculate the following surface area: the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially. My attempt: Let ...
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Simplex like integrals

I am interested in computing the following integral which is somewhat similar to computing volume of simplex in $n-$ dimensions : Let $S_a=\bigg\{(x_1,x_2,\ldots,x_n):\ 0 \leq x_i \leq 1,\ n- \frac{1}...
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369 views

Gradient of surface area element $\nabla . dS$

When finding a surface integral you first must parametrise the surface into $r = r(u,v)$. So the area element is a function of $u$ and $v$. Does that mean that grad of it is zero? That is, is $$\nabla ...
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648 views

Not getting surface integrals

I have this problem from homework: Integrate the given problem over the given surface. $H(x,y,z)=x^2 \sqrt{5-4z}$ over the parabolic dome $z = 1-x^2-y^2, x \ge 0$ I used this formula from my book ...
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How can this be the surface area of an intersection of cone and cylinder?

I have an exercise that requires me to calculate the surface of the intersection between two curves: $Z_1:z^2=x^2+y^2$ and $Z_2:x^2+y^2=2y$. So what I did is the following: Parametrise the cylinder $...
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Change of Variable vs Parametrization

I'm considering the problem of computing the volume and surface area of an ellipsoid. We define the ellipsoid by the set of points $$S=\left\{(\alpha r\cos\theta\sin\phi,\beta r\cos\theta\cos\phi,\...
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Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
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How to set up a surface integral on a cylinder cut by planes?

Find the surface area of the piece of the cylinder $x^2 + y^2 = 4$ cut off by the planes $z = 0$ and $y = z$ with $y \ge 0$ using surface integrals. Can someone help me set up this surface integral ...
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Flux of $\vec F=\langle x^2,-y^2,z^2\rangle$ over region between two parallel planes in the first octant

Use the divergence theorem to compute the net outward flux of $\vec F(x,y,z)=x^2\,\vec\imath-y^2\,\vec\jmath+z^2\,\vec k$ over the region $D$, where $D$ is the space between the planes $z=3-x-y$ and $...
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Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$

Compute the integral $\int_S{\vec{r} \cdot \hat{n}dS}$, where $S$ is the surface of the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$. I have calculated and thoroughly checked each step and the result I am ...
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Volume of the fluid in the upward direction

Let $\mathbf{V}$ the velocity vector of a fluid particle at the point $(x,y,z)$ in a steady-state fluid flow. $$\mathbf{V}=x\hat{\mathbf{i}}+y\hat{\mathbf{j}}+3z\hat{\mathbf{k}}$$ Let $S$ be the ...
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179 views

Flux of a hemisphere

I have been asked to compute the flux through a hemisphere radius 2 centred at the origin oriented downward with $$ \overrightarrow{F}=(y,-x,2z) $$ I have worked out that $$ \hat{n}=-\frac{(x,y,z)}{...
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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 ...
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How to Calculate the flux of the Vector Field on the surface $z = 1-x^2-y^2$ ( getting normal vector $(0,0,0)$ at the point $(0,0,1)$ ?!!! )

Let $S$ be the surface $z = 1-x^2-y^2 , 0\leq z$. Find $\int_{S} x^2z~dydz + y^2z~dzdx + (x^2+y^2)~dxdy$. Choose the direction of the normal upwards. so i calculated the flux and i got that it ...
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How to find the surface integral of Torus intersecting with cylinder ??

Let $T$ be the torus obtained by revolving the circle {$(x,0,z)| (x-3)^2 + z^2 = 1$} about the $z$-axis. Find the area of the surface obtained by taking the intersection of $T$ with the ...
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Applications of Green's theorem

Let $f\in C^1$. Prove that for every $x_*$ : $(\nabla \times f)(x_*)=\lim_{\epsilon \rightarrow 0} \frac{1}{\pi \epsilon ^2} \int_{\partial B_\epsilon (x_*)}f(x)dx$ I know that $$ \lim_{\epsilon \...
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How can we show that the limit of the following surface integral is finite?

I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $\epsilon$ at the ...
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Surface Integral - Mean Value Property

I have been given this question to solve however I'm having some difficulty solving it as I am quite new to Partial Differential Equations: Let $ a = (2,2) $ and $ r =5 $ Compute the following ...
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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 ...
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52 views

Euler's equation

I am studying fluid mechanics from Landau and Lifshitz and I'm trying to prove the equation $\begin{align*}\\ -\oint \; p\,df = -\int\, grad\,p\,dV\end{align*}\\$ (topic on Euler's equation page 2) ...
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108 views

Surface integral of spherical shell in a $e^{-x^2}$ field

I have a density field which is spherically-symmetric. The field is centred at the origin and is parametrized by a radius $R$ and a falloff parameter $k$. If I sample a point whose position vector is $...
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Differentiating function equal to a variable

I'm doing some surface integrals, and sometimes you have a surface given by let's say $z^2=x^2+y^2+1$. From the jacobian for $r = [x,y,f(x,y)]$ we have that it is equal to $[-\frac{\partial f}{\...
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How do I obtain this function for the surface integral to be zero?

I want to find some function $f(y)$ such that $$\int_R[f(y)\ dx + x \cos y \ dy] = 0$$ for all closed contours $R$ in the $(x,y)$ plane. My thoughts: I am thinking of an opposite function something ...
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Help understanding this numerical surface integration technique?

I'm attempting to write a FORTRAN program that calculates the magnetic field, B, at any point outside of a bar magnet. I'm going to use a first order euler scheme, where each side of the bar magnet ...
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186 views

Stoke's Theorem on cylinder-plane intersection.

I was trying the solve the following exercise from Apostol Vol 2: Using Stokes theorem, proof that $\int_{C} (y-z)dx +(z-x)dy+(x-y)dz = 2 {\pi}a(a+b)$, where $C$ is the intersection of the cylinder $x^...
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How is the derivative of a potential with respect to an outward normal equal to the grad of the potential

If we first consider Gauss' Law $$\oint_s \boldsymbol{E\cdot} \,d\boldsymbol{A} = Q_{enclosed\\in\ surface}$$ We know from physics that $\boldsymbol{E}=-\nabla V$, but I want to know is it ...
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129 views

Surface integral of vector field $F(x,y,z)=xz\hat{i}+x\hat{j}+y\hat{k}$ where S is hemisphere

I'm stuck on this problem. I've looked up a bunch of answers and watched youtube videos on solving similar problems, but when I try to apply what I learn to the below problem I get a weird answer ( $...
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20 views

Find the flux across a surface

I came across the following question: Compute the flux out of the hyperboloid $(1) \quad x^2+y^2=1+z^2$ between $z=1$ and $z=-1$ of the force field $\langle x,y,z\rangle$. The way the question is ...