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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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2 answers
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Area of the part of the cylinder $x^2+y^2=2ay$ outside the cone $z^2=x^2+y^2$

Problem: Find the area of the part of the cylinder $x^2+y^2=2ay$ that lies outside the cone $z^2=x^2+y^2$. My attempt: So I thought we could do this by projecting the surface onto the $yz$-plane and ...
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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 ...
1 vote
1 answer
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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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2 answers
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Surface area given a function in polar without converting to rectangular

Note: I need to use the double integral method of integration for this. Let's say I have a function in terms of $r$ and $θ$: $f(r, θ)$ For a rectangular function $f(x, y)$, one can use the following ...
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1 answer
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Find surface area by calculating surface integrals

Fix a radius $r > 0$ and two angles $ϕ_1$ and $ϕ_2$, with $−π/2 < ϕ_1 < ϕ_2 < π/2$ Find the surface area of the portion of the sphere of radius r with latitudes between $ϕ_1$ and $ϕ_2$. ...
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1 answer
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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 ...
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1 answer
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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-...
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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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1 answer
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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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1 answer
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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 ...
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1 answer
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Surface integral over the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and normal making an acute angle with $\vec k$

Let $\vec F=(x^2+y-4,3xy,2xz+z^2)$ and $S$ be the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and $\vec n$ is the unit normal to $S$ making an acute angle with $\vec k$ , then ...
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Finding the volume of 3 dimensional region under the graph of a function.

Im trying to do the following question but im confused. Let W be the three dimensional region under the graph of the function $f(x,y) = \mathrm{e}^{x^2+y^2}$ and over the region in the $(x,y)$ plane ...
1 vote
1 answer
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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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1 answer
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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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1 answer
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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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6 answers
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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}...
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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 $z=\sqrt{a^2-x^2-y^2}$. ...
1 vote
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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 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 ...
1 vote
1 answer
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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 ...
1 vote
1 answer
62 views

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 ...
1 vote
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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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1 answer
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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: $$\int_S(\nabla\otimes\vec{v})\cdot d\vec{S}=\int_S\nabla(\vec{v}\cdot d\vec{S})$$ ...
5 votes
1 answer
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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 ...
1 vote
1 answer
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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 ...
1 vote
0 answers
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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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2 answers
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Parameterization for this surface integral

The problem is: Evaluate the $\int \int_SF* dS$ for the given vector field F and the oriented surface S. for closed surfaces, use the positive (outward) orientation. F(x,y,z) = xi +yj +5k. S is the ...
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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 ...
2 votes
1 answer
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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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2 answers
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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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Use the Stoke's Theorem to evaluate $\oint_CF\cdot dr$, where $F(x,y,z) =(x+y^2)i+(y+z^2)j+(z+x^2)k$

The question on the assignment is: Use the Stoke's Theorem to evaluate $\oint_CF\cdot dr$, where $F(x,y,z) =(x+y^2)i+(y+z^2)j+(z+x^2)k$ and $C$ is a triangle with vertices $(1,0,0)$, $(0,1,0)$, and $(...
2 votes
1 answer
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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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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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Computing the flux integral on the given surface

Compute the flux integral for the field $\mathbf{F} = x\mathbf{i} + y\mathbf{j} + 5\mathbf{k}$, where $S$ is the boundary of the cylinder enclosed by $x^2+z^2=1$ and the planes $y=0$ and $x+y=2$. How ...
3 votes
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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 ...
1 vote
1 answer
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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, ...
2 votes
1 answer
170 views

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 ...
2 votes
1 answer
1k views

Evaluate Surface Integral over this triangular surface

When I solving the practice exercise problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral, but couldn't. Can someone ...
2 votes
1 answer
1k views

Surface integral over a triangle region

Evaluate the integral $\iint xyz\,ds$ where $S$ is the triangle with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Well, I did almost everything, I'm just stuck at finding the boundaries. Firstly, I ...
1 vote
0 answers
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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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