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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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### Tricky surface integral of vector field

The following problem comes from a vector calculus exam. Let $$S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\}$$ be an embedded surface with the orientation ...
18 votes
3 answers
533 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 ...
17 votes
2 answers
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### 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 ...
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13 votes
2 answers
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### 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 ...
• 411
11 votes
2 answers
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### 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 ...
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11 votes
4 answers
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4 answers
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### 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 ...
• 373
7 votes
1 answer
88 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 ...
• 935
7 votes
1 answer
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### 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 ...
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7 votes
0 answers
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### 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 ...
6 votes
1 answer
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• 8,488
6 votes
1 answer
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### 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:...
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6 votes
1 answer
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### Divergence Theorem when Surface isn't closed

So we essentially want to evaluate $$\iint_S \vec{F} \cdot d\vec{S},$$ where $\vec{F} = \langle 2x+y, x^2+y, 3z \rangle$ and $S$ is the cylinder $x^2+y^2=4$, between the surfaces $z=0$ and $z=5$. We ...
6 votes
1 answer
108 views

### Purpose of the "$\vec{F} \cdot \text{d}\vec{S}$" notation in vector field surface integrals

When describing surface integrals in vector fields, it's common to use the notation $$\iint_S \vec{F} \cdot \text{d} \vec{S}$$ as a shorthand for $$\iint_S \vec{F} \cdot \vec{n}\, \text{d}S$$ This ...
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6 votes
2 answers
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### Calculating Surface Area using Differential Forms

I'm trying to reconcile the definition of surface area defined using manifolds vs the classic formula in $\mathbb{R^3}$, but it seems like I'm off by a square. In Spivak's Calculus on Manifolds, the ...
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6 votes
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• 71
6 votes
1 answer
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### 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 ...
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2 answers
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4 votes
2 answers
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### 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) ...
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4 votes
1 answer
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### 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-...
4 votes
2 answers
2k 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
5 answers
137 views

### Computing $\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y$

Computing $$I=\iint_S (y+z)\mathrm{d}y\mathrm{d}z+(z+x)\mathrm{d}z\mathrm{d}x+(x+y)\mathrm{d}x\mathrm{d}y,$$ where $S$ is the upper side of the plane $x+y+z=1$ located inside the interior of the ...
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4 votes
2 answers
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### How to interpret a double integral?

What can be the geometrical meaning of $$\iint_R dA ~~~~~~~~~~ (1)$$ ? It is the particular case of $$\iint_R {f(x,y)} dA$$ where $f(x,y) = 1$. What I have found from my search is that (1) ...
4 votes
3 answers
943 views

### Relationship between $(n - 1)$ forms and flux of a vector field across a hypersurface

I am currently studying about differential forms and want to deduce the Divergence Theorem (the one in $\mathbb{R}^n$) from the general Stokes' Theorem, which is obtained by taking \omega = \sum_{i=...
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4 votes
1 answer
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### 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: ...
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4 votes
1 answer
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### 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 ...
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4 votes
1 answer
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### 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
2 answers
357 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 ...
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