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Questions tagged [stokes-theorem]

Stokes' theorem is a result about integration of differential forms.

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Evaluate $\oint_C F.dr$ where $F = (x-z)i +(2y)j +(x+z)k$

I'm having trouble figuring out $$ \oint_C \underline{F} \cdot d\underline{r}, $$ where $\underline{F} = (x-z)\underline i +2y\underline j +(x+z)\underline k$ and in which $C$ is a closed path in the $...
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Why is it that the surface integral of the flux of a vector field is the same as the surface integral of the vector field itself?

In other words, this: http://www.math.ucla.edu/~archristian/teaching/32b-w17/week-7.pdf Is this just a definition because what we really care about is how much the vectors are "pushing" through the ...
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integral of a k-form over an oriented compact manifold

I see in my course the following theorem: If $\omega$ is an exact k-form over an oriented compact manifold M of dimension $k$, then $\int_M \omega=0$. I don't have a proof of this theorem and I ...
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Flux of curl of a vector field through cross section of a sphere

Suppose $R>0$, $M$ be the sphere centered at the origin with radius $R$, $M=S^2(0,R)$, and $\Gamma=\{(x,y,z)\in M:x+y+z=1\}$. I have to prove that, with the appropriate orientation of $Gamma$: $$\...
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An intuitive explanation for Green theorem and Divergence theorem

As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize. I think I have found a pretty intuitive way to think about the ...
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Prove that the integral of a differential form is zero

Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^n$ such that $[\mathbb{X},\mathbb{Y}]=0$, and let $\Phi_t$ and $\Psi_s$ denote their respective flows. Let $c:[0,1]\times [0,1]\to \...
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Application of Stokes' Theorem with the vector field $K=(-zy,zx,z^2)$ and the surface $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$

I'm asked to evaluate the flux $$\int_{S} rot K d\omega,$$ where $S$ is the region $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$ and $K$ is the vector field $(-zy,zx,z^2)$. So when I evaluate directly the ...
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Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency

I am having trouble to understand what is going on with the Maxwell–Faraday equation: $$\nabla \times E = - \frac{\partial B}{\partial t},$$ where $E$ is the electric firld and $B$ the magnetic field. ...
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What goes wrong with Stokes theorem if a surface is not orientable?

For the Möbius strip parametrized by $\{\sigma(\theta,r)=((1+r\sin(\theta/2))\cos\theta,(1+r\sin(\theta/2))\sin\theta,r\cos(\theta/2))\ \mid \\ (\theta,r)\in A=(0,2\pi)\times(-1/2,1/2) \}$ we get ...
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Verify Stokes' theorem on a hemisphere using a line integral

When solving this question, I wrote $r(t)$ as $\langle\cos t,0,\sin t\rangle$ then I differentiated it and got $dr=\langle-\sin t,0,\cos t\rangle\,dt$, then i substituted in the formula for line ...
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Determining $R$ so that the flux of the F field through S is maximum

For any $R>0$ let the surface $S={(x,y,z):x^2+y^2+(z-1)^2 = R^2; z≥1}$ oriented with the normal whose third coordinate is positive. And let the field $F(x,y,z) = (zx-x cos (z), -zy+ y cos(z), 4-x^...
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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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Volume form on a compact manifold is not exact

I am trying to show that a volume form $\mu$ on a compact manifold $M$ is not exact, i.e. show there is no $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \mu$. My attempt is the following: Suppose,...
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Intuition on Stokes' Theorem

Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int \int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$ I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
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Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
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Stokes' Theorem - Vector Field

I am having problems trying to verify Stokes' theorem (below) as part of a question. $$\iint_{S} \text{curl} \vec F \cdot d\vec S=\oint_{c} \vec F \cdot d\vec r$$ The vector field in question is $\...
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Frightening Stokes Theorem Computation

I came across a rather frightening problem in my problem set and I am confused about it. I need to compute the flux of a field $\vec F = \langle y, x^2, z(x^2-y^3)^7 \cos(e^{xyz}) \rangle$ through ...
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Using Stokes's Theorem to Compute the Circulation of a Vector Field over a Triangle in the Plane

I'm tasked with computing the circulation of the vector field $\vec F = <y^2, z, xy>$ along the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$ with the orientation of the curve following ...
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calculate the surface integral in the upper hemisphere

Calculate the surface integral $f(x,y,z)=x^2+y^2+z^2$ in the upper hemisphere of the sphere $x^2+y^2+(z-1)^2=1$ I tried to compute the value of the surface integral $\iint_S{F.n} dS$ with the ...
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Gauss-bonnet just for geodesic triangles

I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is $$\iint_R KdS+\sum_{i=0}^k\int_{s_i}^{s_{i+...
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Solve the following integral using Stokes Theorem.

I am asked to evaluate the following integral: $$\int\int \text{curl} \ \vec{F} \cdot d\vec{S}$$ where $F = (y,-x,zx^3y^2)$ and $S$ is the surface given by $x^2 + y^2 + 3z^2 = 1$ with $z \leq 0$. I ...
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No where vanishing exact $1$-form on compact manifold.

I found several answers on the following question : Does there exists a no where vanishing exact $1$-form on a compact manifold without boundary? All answer says that certainly not. But I cannot ...
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How do I use Stokes' theorem to find curl

How do I solve questions that ask me to use Stokes' Theorem to find curl F For example: Use Stoke's Theorem to find curl F: $$F(x, y, z) = \langle e^x+y^2, y^2+z^2, \sin(z)+x^2\rangle$$ $\iint_S ...
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Differential Forms and Applications by do Carmo - Divergence theorem

I read the proof of Stokes theorem for manifolds by do Carmo's book and I'm trying understand an example (the Divergence theorem) given after the proof of Stoke's theorem, but I didn't understand. A ...
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Why is curl considered the differential operator in 3-space?

Why is the curl considered the differential operator in 3-space instead of the gradient? It would seem that the gradient is the corollary to the derivative in 2-space when extending to 3-space. This ...
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Orientation in the proof of Stokes Theorem

I'm reading the proof of Stokes theorem at page 83 of "Godinho, Natàrio, An introduction to Riemannian geometry" and I can't understand a passage in it, probably because the definition of ...
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Stokes First Problem/Rayleigh with a wall separation of variables

like the title says, I would like to calculate Stokes first problem with a fixed wall at d. We have to do it with separation of variables. If I take the obvious Ansatz, $\lambda=0$ my time dependecy ...
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Understanding the Difference between vector and velocity fields

I'm having difficulty understanding the difference between a vector and velocity field. In particular, there are two questions which I had to complete which ask to find the flux of a velocity field. ...
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Verifying Stokes' Theorem for an upper hemisphere

There is a hemisphere, radius $1$, centred at $(0,0,0)$, where the vector field is $$\vec F = \Big(x^3+\frac{z^4}{4}\Big) \hat i + 4x \hat j + (xz^3+z^2) \hat k$$ Verify Stokes' theorem for this ...
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Calculating Flux of a surface

I have to calculate the flux a surface but I don't really find the way to parametrize the surface. Moreover, I am not sure if I have to use Gauss theorem or Stoke's theorem. This is my exercise : ...
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The proof of Riemann’s bilinear relations

I am reading the proof from http://page.math.tu-berlin.de/~bobenko/papers/2011_Bob.pdf I am very confused the step: (from Stokes theorem) $$\int_Rw\wedge w'=\int_{\partial F_g}w'(p)\int_{P_0}^Pw$$ ...
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Using Stoke's Theorem on a 2D vector field?

So i'm supposed to calculate the line integral $$\int_C\mathbf{F}\cdot d\mathbf{l}$$ where $\mathbf{F}=(xy^2+2y)\vec{\mathbf{x}}+(x^2y+2x)\vec{\mathbf{y}}$ Through curve $C_1$ which contains two ...
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Clarifying Stokes' Theorem misconceptions with a problem

Question: Using Stokes's Theorem for $A=(y+z)\hat i-(xz) \hat j+(y^2)\hat k$ where $S$ is the surface of the region in the first octant bounded by $2x+z=6$ and $y=2$ which is not included in the (a) $...
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Verify Stokes' Theorem for $v=zi+xj+yk$ over the hemispherical surface $x^2+y^2+z^2=1$ and $z \gt 0$.

Verify Stokes theorem if $v=zi+xj+yk$ (where $i,j,k$ are the identity vectors for the $x,y,z$ axis) is taken over the hemispherical surface $x^2+y^2+z^2=1$ and $z \gt 0$. Stokes theorem being: $$\int\...
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Help with Green's theorem

I am struggling doing a calculation with Green's theorem. I have to evaluate this integral $ \int D \mathbf{r}^2 \nabla^2 p(\mathbf{r}, t) d \mathbf{r} $ ($ D$ is a numerical coefficient). Actually I ...
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Simple stokes - why is this the correct orientation?

We want to calculate $\iint_{S}\text{curl}(\vec F)dS$ where $\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)$ and $S$ is the part of the paraboloid $z=x^2+y^2$ that's inside the cylinder $x^2+y^2= 1$ with an outward ...
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Projection of the surface on another surface using Stokes theorem?

Can multiple surfaces connected together be used to project them onto another surface? For example let's take a cube. I have seen different examples from books where they project the upper surface ...
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Stokes' theorem for a flat surface

Let $\vec{F}=\langle x^2,y^2,-z\rangle$. I want calculate the line integral of $F$ on curve $C$ which is a closed loop consisting of the line segments that form the triangle with vertices $(0,0,0), (0,...
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verifying Stokes' theorem 4

Verify the Stokes' theorem for the function $\mathbf F = x \mathbf i + z \mathbf j + 2 y \mathbf k$, where $\mathcal{C}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $...
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1answer
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On line integrals of $\frac{xdy-ydx}{x^2 +y^2}$

There are a few questions on MSE about integrals of the form $$\int_C \frac{xdy-ydx}{x^2 +y^2},$$ where $C$ is a smooth simple closed positively oriented curve; but none of them gave me a complete ...
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Evaluate Integral Using Stokes Theorem

Using Stokes theorem evaluate $$\int_\Gamma(z-y)\,dx-(x+z)\,dy-(x+y)\,dz$$ where $\Gamma$ is the intersection of $x^2+y^2+z^2=4$ with the plane $z=y$ with anticlockwise orientation when looking on $...
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Stokes theorem, intersection between cylinder and plane

$\mathcal C$ is the intersection curve between the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$. I tried parameterizing the curve by expanding the cylinder equation $x^2 + (y-1)^2 = 1$. I think I ...
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1answer
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Evaluate $\int_M(x-y^2+z^3)dS$

Evaluate $\int_M(x-y^2+z^3)dS$ when $M$ is the part of the cylinder $x^2+y^2=a^2$ where $a>0$ which is between the two planes $x-z=0$ and $x+z=0$. So I did not manage to use green/gauss/stocks, so ...
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Given rotor and curve find circulation of a vector field

Given the curve $C$ of equation $$\vec X=(3\cos t,3\sin t,6\cos t),\qquad0\leq t\leq2\pi$$ oriented according imposes this parameterization, find the circulation of $\vec f$ along $C$ if $\vec f\in\...
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Given the union of a cylinder and a sphere calculate the line integral using Stoke's theorem

Given the surface $S=S_1\cup S_2$ where $S_1=\{ (x,y,z) \in \mathbb{R}^3 / x^2+y^2=1 ; \space 0 \leq z\leq1 \}$ and $S_2=\{ (x,y,z) \in \mathbb{R}^3 / x^2+y^2+(z-1)^2=1 ; \space z\geq1 \}$, ...
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Exercise about Stokes with alternative solution

Exercise text Calculate $ \int_\gamma ydx-xdy+z^2dz $ where $ \gamma $ is the closed curve generated by the intersection of $ x^2+y^2=4 $ with $ z=y^2 $. My solution and question I tried to solve ...
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Understanding the use of Stokes' Theorem

Transform the surface integral $\int_S \text{rot}\vec{F}\cdot \vec{dS}$ in a line integral using the Stokes Theorem and then calculate the line integral for: $\vec{F}(x,y,z)=(y,z,x)$, where $S$ is the ...
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Combinatorial analog of holonomy on a planar graph with quadrilateral faces

The concept of holonomy comes from differential geometry. It describes the behaviour of a vector on a surface when it is moved via parallel transport along a closed curve on the surface. A similar ...
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Find the following parameterizations for both the surface S

Let S be the portion of the plane $ \ 2x + 3y + z = 4 \ $ lying between the points $ (−1, 1, 3), (2, 1, −3), (2, 3, −9), (−1, 3, −3) $. Find the following parameterizations for both the surface S ...
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Green's Theorem and the Cauchy Integral Formula/Cauchy's Theorem

So we know by Green's Theorem that for $F: \mathbb{R}^2 \to \mathbb{R}^2$, $F = (P,Q)$, with $F$ defined on some open set $U$ such that $\partial U$ is piecewise smooth and simple (see JCT) that we ...