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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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Calculating a surface integral of a surface by another surface

Let $\Sigma$ be the surface $\frac{x^2}{2}+y^2=1$, $0\leq z\leq 1$ and $u(x,y,z)=\frac{x}{x^2+y^2},\frac{y}{x^2+y^2},z^2-z)$. Calculate the flux integral $\iint_\Sigma (u|N)dS$ where $N$ points away ...
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How to find area of circle using double integral?

I was trying to verify Stokes' theorem for a function and this is where I'm confused. $$\nabla \times \mathbf{v} = \mathbf{ k} \equiv\text{ct. vector}$$ So, $$\displaystyle\oint (\nabla \times\mathbf{...
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Integral of log-concave function on the boundaries of convex sets

Let $g$ be a convex function on $\mathbb{R}^d$ and $B_1 \subset B_2 \subset \mathbb{R}^d$ be two convex sets with smooth boundaries $\partial B_1, \partial B_2$. Then do we always have the following ...
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Computing a surface integral

I have to verify that the line integral and the surface integral of Stokes' Theorem are equal for the vector field $\boldsymbol{\mathrm{F}}(x,y,z)=(x,y,z)$ and the portion of the surface $S$ defined ...
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What is the surface integral?

Let $u=((x+y)z,y+xz,z^2+xz)$ be a vector field and $\Sigma$ be the surface $\frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}=1$. Calculate $\iint_{\Sigma}(u\cdot N)dS$. So $\text{div }u=1+3z+x$ and by using ...
per persson's user avatar
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Derivation of the formula for parametrized surface area element

The surface integral is given by $$\int_{S} f \,dS = \iint_{D} f(\mathbf{\sigma}(u, v)) \left\|{\partial \mathbf{\sigma} \over \partial u}\times {\partial \mathbf{\sigma} \over \partial v}\right\| \,...
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Calculating Electric Flux Through a Closed Surface

I'm trying to solve a problem involving the calculation of electric flux through a closed surface, but it's my first time attempting such a problem and I could use some guidance. Any help would be ...
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The surface integral $\iint_S (z^2 + y^2 + x^2) \, dS$ over the cube $S$? [closed]

Evaluate the integral $$\iint_S (z^2 + y^2 + x^2) \, dS ,$$ where $S$ is the surface of the cube $\{-a < x < a, -a < y< a, -a< z< a\}$. I've attempted to partition the surface into ...
Алина Давыдова's user avatar
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Restriction of a compactly supported function on a bounded domain in a surface

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\...
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How do i prove that the these two definitions of divergence are equivalent?

In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal ...
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How do I parameterize this double integral?

I am new to multivariable calculus and I am trying to learn how surface area double integrals work. I am stuck on how to parameterize this function: $\iint_S x dS$ where $S$ is the part of the ...
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Question About Orientation and Surface Level Integrals

I'm stuck on this question: Let S be a level surface defined by f (x, y, z) = k where k is a constant. Show that the two possible orientations of S can be chosen as ± ∇f / ‖∇f ‖ I believe in order to ...
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Evaluate integral of $\langle xz^2,x^2y-z^3,2xy+y^2z\rangle$ with divergence theorem

Q. Evaluate by Gauss divergence theorem $$\iint_S xz^2\,dy\,dz + \left(x^2y-z^3\right)\,dz\,dx + \left(2xy+y^2z\right)\,dx\,dy$$ where $S$ is the surface bounded by $z=0$ and $z=\sqrt{a^2-x^2-y^2}$. ...
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Fourier transform as an integral on a surface

Let $v\in C_c^\infty(\mathbb{R}^n)$ with support $B\subset \mathbb{R}^n.$ Let $h$ be real valued smooth function on a neighborhood of $B$ and $\xi \in \mathbb{R}^n, t\in \mathbb{R}$ and consider the ...
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How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?

How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously? This equation is called “Gauss's Law” in physics. I seek for a rigorous mathematical proof for ...
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How to properly (rigorously?) treat a dot product integral?

I have a very long cylindrical mass. Now say I have some cylindrical Gaussian surface $\Omega$ centered around this cylinder, visualized in purple below (ignore everything else in the diagram), for ...
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Surface Integral of vector field bounded by two spheres .

enter image description here A vector field : $$D\mapsto \mathbf{a}_r\frac{\cos^2\phi}{r^3}+\mathbf{a}_\theta\sin\theta$$ Exists in the region between two spherical shells defined by r=1 cm and r=2 cm ...
mohanad shalalfeh's user avatar
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Seeking suggestions for a book with hard problems about surface and volume integrals

I am interested about the hard problem of surface and volume integral, so can anyone suggest me a book based on the problem on surface and volume integral (containing a lot of hard problem) for ...
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Stokes Theorem for the intersection of $z=x^2+y^2$ and $z=x+2$ and parameterizing the portion of the plane inside of $z=x^2+y^2$

I have been trying to solve the surface integral $$\int \int_S curl(\vec{F}) \cdot \vec{dS}$$ using stokes theorem for the vector field $$F(x,y,z)=\left(xz,yz,xy\right)$$ and where $S$ is the ...
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Confirmation of calculation: $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})dS$

The question goes like this: Evaluate $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})\,dS$ where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ in the first octant. I saw ...
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Why is Hausdorff Measure used on surfaces integrals?

I have a mild curiosity as to why are we using the Hausdorff measure to define surface integrals (for example the co area formula) and not use instead the Lebesgue measure, or at least on my class and ...
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Gradient and differential surface element

In my textbook There is the following step made when dealing with a certaion surface integral over a random surface: $$\int_S(\nabla\otimes\vec{v})\cdot d\vec{S}=\int_S\nabla(\vec{v}\cdot d\vec{S})$$ ...
Krum Kutsarov's user avatar
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Calculate the surface integral $\int\int_S(x+y^2)d\sigma$

The problem: Calculate the surface integral $\int\int_S(x+y^2)d\sigma$ where $S$ is the part of the surface $x^2+y^2=4$ that is between the planes $z=0$ and $z=3$ What I've tried: I described the ...
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Surface integral of given vector field through a given surface

(Problem) Evaluate the surface integral, where $\vec{F}(x,y,z)=(x^2-y^2+y)\vec{i}+2xy\vec{j}+(y^2-z^2)\vec{k}$ and $S$ is the surface of the cylinder $x^2+y^2=1,0 \leqq z \leqq 1$ I solved this ...
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Fourier series of surface in terms of spherical harmonics

During my studies I came across the following integral $$\oint_S\mathrm{d}S\,\mathrm{e}^{-\mathrm{i}\vec{x}\cdot\vec{k}},$$ where $S$ is a 2D surface in 3D space, and noticed that it almost looks like ...
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What exactly does "outward normal vector" means when talking about an orientation?

I am asked to calculate the integral of a $2$-form on the sphere: $$ \int_{S^2} \frac{x\mathrm{d}y\wedge \mathrm{d}z-y\mathrm{d}x\wedge \mathrm{d}z+z\mathrm{d}x\wedge \mathrm{d}y}{(x^2+y^2+z^2)^{\frac{...
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Find flux of $F=\frac{(x,y,z)}{(x^2+y^2+z^2)^{3/2}}$ in surface.

Let $\vec{F}=\left(\frac{x}{(x^2+y^2+z^2)^{\frac32}},\frac{y}{(x^2+y^2+z^2)^{\frac32}},\frac{z}{(x^2+y^2+z^2)^{\frac32}}\right)$ be a vector field. Denote $A=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2\leq 3, ...
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Surface integral below $z=9-x^2-y^2$ but above $z=5$

I have to evaluate $$\iint_{S}\vec{F}\cdot d\vec{S}$$ with $$\vec{F}(x,y,z)=(y,-x,z)$$ and $S$ is bounded by $z=9-x^2-y^2$ and $z=5$. I can't use Gauss Theorem to do it. I thought about using the ...
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Mean Value Theorem for Surface Integrals

Let $F$ be a continuous vector field; then: $$\iint_{S}F d\vec s=F(Q)n(Q)A(S)$$ for some $Q \in S$ and $A(S)$ is the area of $S$. Firstly, I arrived at $\iint_{D}fg dA=f(Q) \iint_{D}g$ dA if $g \geq ...
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What is wrong in this derivation of the time derivative of a flux?

There are several resources, including this question as well as for example problem 5-1 in Kovetz "The Principles of Electromagnetic Theory" which state that the time derivative of a flux ...
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Surface Integral Over Part of Plane

Let $S$ be the surface where $x + 2y + 3z = 0$ and $-1 \leq y \leq 1, 0 \leq z \leq 1$. Compute the surface integral $$\int \int_S (2x,3y-x,1-2y) \cdot \mathbf{\hat{N}}dS$$ where the unit normal ...
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Divergence's Theorem Application

Let $\Omega \subset \mathbb{R}^3$ be the solid given by $$\left\{(x,y,z) \in \mathbb{R}^3:x^2+y^2+z^2 \leq 1; x + 2y − z \geq 0\right\}$$ and $N$ the unitary normal vector field to the surface $\...
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Struggling with a Calculus III problem :$\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
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Computing Surface Integral over Tetrahedron

Given a vector field $\mathbf{F}(x,y,z) = (2xz,0,(x-1)^2)$ and three points $P_1 = (1,0,0)$, $P_2 = (0,1,0)$, $P_3 = (0,0,2)$ in $\mathbb{R^3}$. Let $T$ be the tetrahedron with corners in $P_1, P_2, ...
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Computing the surface integral of $\ G(x,y,z) = xyz $ over the triangular surface with vertices $(1,0,0),\,(0,2,0)$ and $(0,1,1)$.

I was attempting to compute the surface integral of $\ G(x,y,z) = \ xyz $ over the triangular surface with vertices at $\ (1,0,0)$, $\ (0,2,0)$ and $\ (0,1,1)$. Clearly, the first step is to ...
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On elementary proof of Stokes' theorem

I have looked at several elementary proofs (i.e., using only basic calculus, and not using differential form or manifold) of Stokes' theorem in books and Wikipedia, and all seem to use the fact that ...
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Lebesgue differentiation theorem extended for surface integrals

Theorem 6, pg 649 in appendix E of evans states the Lebesgue differentiation theorem: THEOREM 6 (Lebesgue's Differentiation Theorem). Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be locally summable. ...
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Area of a surface by means of a double integral

Considering the surface in $\mathbb{R^3}$ given by: $x^2-y^2=1$; $x>0;-1<y<1;0\leq z \leq 1$ Calculate its area by a double integral via a parametrization of the surface. Firsly I setted ...
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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 ...
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Using Gauss theorem to evaluate the flux

Given $$F(x,y,z)=(x^3+\sin(z),x^2y+\cos(z),\exp(x^2+y^2))$$ I have to evaluate the flux of $F$ through $S$, with $S$ being the surface of $Q$, such that $Q$ is bounded by the cylinder $$z=4-x^2$$ the ...
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Surface Integral over Cylinder with peculiar bounds

Been struggling with this problem for a while and can't seem to get the bounds right. The problem: Evaluate the surface integral $$ \int\int_S \vec{F} \cdot d\vec{S} $$ $$\vec{F}(x,y,z) = x\vec{i} + y\...
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Change of basis and multivariate Taylor series in surface integral

Let $\Gamma$ be a closed smooth surface in $\mathbb{R}^{3}$, and $\mu:\Gamma\rightarrow\mathbb{R}$. We assume $\Gamma$ can be parametrized in $(u,v)$ such that $\mathbf{x}(u,v)\in\Gamma$ for $(u,v)\in ...
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How to solve this kind of surface integral with Hamilton Operator?

In $\mathbb{R}^3$, $f=\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2+\left(\frac{z}{4}\right)^2$, Surface $S$ is defined by $S=\{(x,y,z)|f(x,y,z)=1, z>0\}$, and the vector field $A$ is ...
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How to calculate the limits of the integral and the outward vector? How to make change of variables in this proof?

I would like to see the calculations behind this proof I found here: Proof of polar coordinates theorem in Evans' PDE Book That is, I am struggling to understand the following: How do we ...
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What is the parametrization of this tetrahedron so that it satisfies assumption in Stokes' theorem?

This question is about a problem in Apostol's Calculus, Vol II, section 12.13 about Stokes' theorem. A question about this problem has been asked before, but that question is about solving the ...
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Find the main part of $\frac{1}{4\pi R^2}\iint_{\Sigma}u(x,y,z)dS-u(u_0,y_0,z_0)$.

Let $u(x,y,z)$ be a continuous function, it has continuous second order partial derivatives at $M(x_0,y_0,z_0)$. $\Sigma$ is a sphere centered at $M$ with radius $R$, and $$T(R)=\frac{1}{4\pi R^2}\...
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Calculation of Surface Integration.

I've been studying surface integration by myself, but I'm always stuck at the last step. Consider the above question: This is my approach: Calculation of the curl of the given field. Calculation of ...
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Change of variables method for surface integrals

Let $\phi: \partial B_1(0) \to \mathbb{R}$ be a differentiable function where $\partial B_1(0) = \{x \in \mathbb{R}^n : |x| = 1\}$ . Defining: $$S = \{\phi(x)x : x\in \partial B_1(0)\}$$ I am ...
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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 $...
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Demonstration of an identity between a simple and a double integral of a Gaussian-Lorentzian product

In the context of a physics problem, I found that a double integral of Gaussian and Lorentzian products has values that are extremely close to that of a single integral. By defining $D(\sigma) = \frac{...
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