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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1answer
26 views
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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1answer
20 views
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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0answers
10 views
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 $Gamma$:
$$\...
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0answers
31 views
An intuitive explanation for Green theorem and Divergence theorem
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the ...
1
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1answer
35 views
Prove that the integral of a differential form is zero
Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^n$ such that $[\mathbb{X},\mathbb{Y}]=0$, and let $\Phi_t$ and $\Psi_s$ denote their respective flows. Let $c:[0,1]\times [0,1]\to \...
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2answers
73 views
Application of Stokes' Theorem with the vector field $K=(-zy,zx,z^2)$ and the surface $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$
I'm asked to evaluate the flux $$\int_{S} rot K d\omega,$$ where $S$ is the region $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$ and $K$ is the vector field $(-zy,zx,z^2)$.
So when I evaluate directly the ...
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2answers
134 views
Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency
I am having trouble to understand what is going on with the Maxwell–Faraday equation:
$$\nabla \times E = - \frac{\partial B}{\partial t},$$
where $E$ is the electric firld and $B$ the magnetic field. ...
2
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1answer
54 views
What goes wrong with Stokes theorem if a surface is not orientable?
For the Möbius strip parametrized by $\{\sigma(\theta,r)=((1+r\sin(\theta/2))\cos\theta,(1+r\sin(\theta/2))\sin\theta,r\cos(\theta/2))\ \mid \\ (\theta,r)\in A=(0,2\pi)\times(-1/2,1/2) \}$
we get ...
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1answer
42 views
Verify Stokes' theorem on a hemisphere using a line integral
When solving this question, I wrote $r(t)$ as $\langle\cos t,0,\sin t\rangle$ then I differentiated it and got $dr=\langle-\sin t,0,\cos t\rangle\,dt$, then i substituted in the formula for line ...
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1answer
26 views
Determining $R$ so that the flux of the F field through S is maximum
For any $R>0$ let the surface $S={(x,y,z):x^2+y^2+(z-1)^2 = R^2; z≥1}$ oriented with the normal whose third coordinate is positive. And let the field $F(x,y,z) = (zx-x cos (z), -zy+ y cos(z), 4-x^...
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1answer
88 views
Where is my mistake? Calculating surface integral/Stoke's theorem
Let
$F(x,y,z)= \begin{pmatrix} -y \\ 2x\\z \end{pmatrix}$
be a vector field and $A$ a hemisphere
with $x^2+y^2+z^2=9 $ , $ z>0 $
with a circular edge at the $x,y $- level with the unit normal ...
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1answer
41 views
Volume form on a compact manifold is not exact
I am trying to show that a volume form $\mu$ on a compact manifold $M$ is not exact, i.e. show there is no $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \mu$.
My attempt is the following: Suppose,...
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0answers
43 views
Intuition on Stokes' Theorem
Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int
\int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$
I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
0
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0answers
18 views
Integration of a (0,1)-form on the boundary of a Riemann surface
In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
2
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1answer
59 views
Stokes' Theorem - Vector Field
I am having problems trying to verify Stokes' theorem (below) as part of a question.
$$\iint_{S} \text{curl} \vec F \cdot d\vec S=\oint_{c} \vec F \cdot d\vec r$$
The vector field in question is $\...
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0answers
63 views
Frightening Stokes Theorem Computation
I came across a rather frightening problem in my problem set and I am confused about it.
I need to compute the flux of a field $\vec F = \langle y, x^2, z(x^2-y^3)^7 \cos(e^{xyz}) \rangle$ through ...
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1answer
38 views
Using Stokes's Theorem to Compute the Circulation of a Vector Field over a Triangle in the Plane
I'm tasked with computing the circulation of the vector field $\vec F = <y^2, z, xy>$ along the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$ with the orientation of the curve following ...
0
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1answer
36 views
calculate the surface integral in the upper hemisphere
Calculate the surface integral $f(x,y,z)=x^2+y^2+z^2$ in the upper hemisphere of the sphere $x^2+y^2+(z-1)^2=1$
I tried to compute the value of the surface integral $\iint_S{F.n} dS$ with the ...
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0answers
29 views
Gauss-bonnet just for geodesic triangles
I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is
$$\iint_R KdS+\sum_{i=0}^k\int_{s_i}^{s_{i+...
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0answers
46 views
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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votes
3answers
77 views
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 ...
1
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1answer
84 views
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 ...
0
votes
1answer
57 views
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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1answer
59 views
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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1answer
65 views
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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0answers
14 views
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 ...
2
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0answers
41 views
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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1answer
81 views
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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1answer
54 views
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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1answer
38 views
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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1answer
56 views
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 ...
2
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0answers
45 views
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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2answers
115 views
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:
$$\int\...
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0answers
27 views
Help with Green's theorem
I am struggling doing a calculation with Green's theorem. I have to evaluate this integral
$ \int D \mathbf{r}^2 \nabla^2 p(\mathbf{r}, t) d \mathbf{r} $ ($ D$ is a numerical coefficient). Actually I ...
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2answers
27 views
Simple stokes - why is this the correct orientation?
We want to calculate $\iint_{S}\text{curl}(\vec F)dS$ where $\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)$ and $S$ is the part of the paraboloid $z=x^2+y^2$ that's inside the cylinder $x^2+y^2= 1$ with an outward ...
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0answers
28 views
Projection of the surface on another surface using Stokes theorem?
Can multiple surfaces connected together be used to project them onto another surface?
For example let's take a cube.
I have seen different examples from books where they project the upper surface ...
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1answer
48 views
Stokes' theorem for a flat surface
Let $\vec{F}=\langle x^2,y^2,-z\rangle$. I want calculate the line integral of $F$ on curve $C$ which is a closed loop consisting of the line segments that form the triangle with vertices $(0,0,0), (0,...
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1answer
50 views
verifying Stokes' theorem 4
Verify the Stokes' theorem for the function $\mathbf F = x \mathbf i + z \mathbf j + 2 y \mathbf k$, where $\mathcal{C}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $...
2
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1answer
101 views
On line integrals of $\frac{xdy-ydx}{x^2 +y^2}$
There are a few questions on MSE about integrals of the form $$\int_C \frac{xdy-ydx}{x^2 +y^2},$$ where $C$ is a smooth simple closed positively oriented curve; but none of them gave me a complete ...
2
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2answers
47 views
Evaluate Integral Using Stokes Theorem
Using Stokes theorem evaluate
$$\int_\Gamma(z-y)\,dx-(x+z)\,dy-(x+y)\,dz$$
where $\Gamma$ is the intersection of $x^2+y^2+z^2=4$ with the plane $z=y$ with anticlockwise orientation when looking on $...
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2answers
55 views
Stokes theorem, intersection between cylinder and plane
$\mathcal C$ is the intersection curve between the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$. I tried parameterizing the curve by expanding the cylinder equation $x^2 + (y-1)^2 = 1$. I think I ...
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1answer
52 views
Evaluate $\int_M(x-y^2+z^3)dS$
Evaluate $\int_M(x-y^2+z^3)dS$ when $M$ is the part of the cylinder $x^2+y^2=a^2$ where $a>0$ which is between the two planes $x-z=0$ and $x+z=0$.
So I did not manage to use green/gauss/stocks, so ...
0
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1answer
27 views
Given rotor and curve find circulation of a vector field
Given the curve $C$ of equation $$\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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1answer
30 views
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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0answers
113 views
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 ...
0
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1answer
50 views
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 ...
2
votes
0answers
22 views
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 ...
0
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0answers
12 views
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 ...
1
vote
2answers
154 views
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 ...
