Questions tagged [stokes-theorem]
Stokes' theorem is a result about integration of differential forms.
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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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3answers
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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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1answer
60 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 ...
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1answer
48 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
51 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
48 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
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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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0answers
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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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1answer
34 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
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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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1answer
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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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1answer
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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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0answers
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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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2answers
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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:
$$\int\...
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0answers
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Help with Green's theorem
I am struggling doing a calculation with Green's theorem. I have to evaluate this integral
$ \int D \mathbf{r}^2 \nabla^2 p(\mathbf{r}, t) d \mathbf{r} $ ($ D$ is a numerical coefficient). Actually I ...
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2answers
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Simple stokes - why is this the correct orientation?
We want to calculate $\iint_{S}\text{curl}(\vec F)dS$ where $\vec{F}(x,y,z)=(y^2z, xz,x^2y^2)$ and $S$ is the part of the paraboloid $z=x^2+y^2$ that's inside the cylinder $x^2+y^2= 1$ with an outward ...
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0answers
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Projection of the surface on another surface using Stokes theorem?
Can multiple surfaces connected together be used to project them onto another surface?
For example let's take a cube.
I have seen different examples from books where they project the upper surface ...
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1answer
39 views
Stokes' theorem for a flat surface
Let $\vec{F}=\langle x^2,y^2,-z\rangle$. I want calculate the line integral of $F$ on curve $C$ which is a closed loop consisting of the line segments that form the triangle with vertices $(0,0,0), (0,...
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1answer
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verifying Stokes' theorem 4
Verify the Stokes' theorem for the function $\mathbf F = x \mathbf i + z \mathbf j + 2 y \mathbf k$, where $\mathcal{C}$ is the curve obtained by the intersection of the plane $z=x$ and the cylinder $...
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1answer
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On line integrals of $\frac{xdy-ydx}{x^2 +y^2}$
There are a few questions on MSE about integrals of the form $$\int_C \frac{xdy-ydx}{x^2 +y^2},$$ where $C$ is a smooth simple closed positively oriented curve; but none of them gave me a complete ...
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2answers
44 views
Evaluate Integral Using Stokes Theorem
Using Stokes theorem evaluate
$$\int_\Gamma(z-y)\,dx-(x+z)\,dy-(x+y)\,dz$$
where $\Gamma$ is the intersection of $x^2+y^2+z^2=4$ with the plane $z=y$ with anticlockwise orientation when looking on $...
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2answers
45 views
Stokes theorem, intersection between cylinder and plane
$\mathcal C$ is the intersection curve between the cylinder $x^2 + y^2 = 2y$ and the plane $y = z$. I tried parameterizing the curve by expanding the cylinder equation $x^2 + (y-1)^2 = 1$. I think I ...
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1answer
51 views
Evaluate $\int_M(x-y^2+z^3)dS$
Evaluate $\int_M(x-y^2+z^3)dS$ when $M$ is the part of the cylinder $x^2+y^2=a^2$ where $a>0$ which is between the two planes $x-z=0$ and $x+z=0$.
So I did not manage to use green/gauss/stocks, so ...
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1answer
25 views
Given rotor and curve find circulation of a vector field
Given the curve $C$ of equation $$\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
28 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
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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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1answer
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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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0answers
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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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0answers
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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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2answers
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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 ...
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1answer
45 views
Determine the orientation of a normal vector for Stokes' Theorem
Use Stokes's Theorem to show that
$$\oint_C=y\,dx + z\,dy + x\,dz = \sqrt{3}\pi a^2,$$ where $C$ is the
suitably oriented intersection of the surfaces $x^2+y^2+z^2=a^2$ and
$x+y+z=0$.
We get ...
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0answers
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Is this equality true? (cauchy integral + green's theorem)
Is this just TRUE?
$$\oint_{\partial S} f(z) \, dz = i\iint_S \bigg[\frac{\partial{f}}{\partial{x}}+i \frac{\partial{f}}{\partial{y}}\bigg] \, dx \, dy$$
Because I'm using it without a proof, and I ...
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0answers
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Apllying Generalized Stokes Theorem
I'm having no difficulties to understand the theorical concepts about the Generalized Stokes Theorem, but I'm having trouble to applying it.
Let $M$ the semiellipsoid in $\mathbb{R}^{3}$ defined by $...
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1answer
39 views
Stokes theorem and integration over fibers
I want to understand the following:
Let $\pi:X \to Y$ a fiber bundle and $\omega$ a closed smooth differential form. Define $I: Y\to \mathbb C$, $y\mapsto I(y)=\int_{\pi^{-1}(y)} \omega$. Then Stoke'...
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1answer
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Tried a line integral and then did it using Stokes theorem, but I'm getting different results…
I'm getting two different results by using two methods that should be equivalent...
"Find the line integral $\int x.dx+y.dy+(x^2+y^2).dz$ along the curve C. C is the intersection between paraboloid $...
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1answer
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Multivariable Calculus. Help o how to integrate and To prove a Rotational property
I've got a couple of doubts in the following excercises. Sorry if somethings wrong or hard to understand in my writing, I'm non-native speaker.
Calculate de integral of the function:
$$f(x,y) = \...
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0answers
39 views
Integration on Sphere and Stoke's Theorem
I would like to clarify the not quite correct statement that one hears every now and then
$$
\tag{1}\label{1}
\color{red}{\text{"The sphere has no boundary, and the integrand is a derivative so the ...
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1answer
72 views
Stokes Theorem orientation
A weird thing about Stokes’ theorem is that if $S_1$ and $S_2$ are two oriented surfaces in $R^3$
that share the
same boundary $∂S$ with the same induced orientation on $∂S$, then $\iint_{S_1}
\...
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0answers
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Generalization of Gauss-Stokes theorem
The Gauss-Stokes theorem defined for a vector field is as follows:
$$
\int_V A^\alpha_{;\alpha}\sqrt{-g}\,d^4x =
\oint_{\partial V} A^\alpha\,d \Sigma_\alpha\, .
$$
But how do we generalize it to ...
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0answers
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Circulation of a vector field.
I need a compute the circulation of a vector field $\overline a = z^2 \overline i + x^2 \overline j + y^2\overline k$ along a curve $\gamma = [x+y+z=1, x^2+y^2+z^2=1]$.
I used a Stokes formula:
$\...
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1answer
51 views
proving Stokes' theorem with the curl
How does a curl explain why the Stokes' Theorem is true? From my understanding, the curl is a circulation of vectors and can be viewed as an infinitesimal circulation to explain it, but I'm confused ...
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1answer
50 views
On Kelvin-Stokes proof without differential forms
I was reading a proof of the Kelvin-Stokes theorem (without differential forms) and the first step was defining a Jordan curve $\gamma:[a,b]\rightarrow\mathbb{R}^2$ and a surface $\psi:D\rightarrow\...
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1answer
26 views
Vector Calculus Proof of Integration
Let R be a solid region bounded by an oriented closed suface $\partial{R}$. Let f and g be $c^2$ scalar functions. Let the $\hat{n}$ be the outward normal to $\partial{R}$. Prove that
$$\iiint \nabla{...
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1answer
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Volume Loss Rate
A spherical hot-air balloon with radius $R$ lies on the $(x_1,x_2)$-Plane.
The balloon has an opening at the bottom right where the radius is $\frac{R}{4}$.
The center of the opening is at the origin....
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2answers
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Proof Line Integral is zero avoiding Stokes
Give is: $C$ which is a closed curve which forms the surface $\Sigma$., $\vec{v} $ which is a constant vector.
I should prove the following expression without using Stokes' Theorem:
$$\oint_C \...
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0answers
28 views
Stokes theorem with two boundaries
The surface $S$ is given by $z=4-x^2-y^2$ for $0 \leq z \leq 3$
and the vectorfield $G(x,y,z) = [x, y, x\sqrt{z^2+1}]$
I want to calculate $$\iint_S curl G*\hat{N} dS$$ where $\hat{N}$ is the unit ...
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0answers
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Multivariable Calculus: Stokes Theorem
Verify Stokes' Theorem for the vector field ${\bf F}(x, y, z)= y{\bf i} +x^2{\bf j}+z{\bf k}$ and the part of the paraboloid $z = x^2+y^2$ that lies below the plane $z=1$
Can anyone please verify if ...
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0answers
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Why is the velocity potential (in fluids) multi-valued in non-simply connected domains?
In Fluid Mechanics, you can define a velocity potential as a closed line integral in a vector field, which equals the sum of curl (vorticity) inside the line over which you are integrating (by Stokes' ...
2
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0answers
118 views
Using Stokes' Theorem on a cube
If we have a cube Q in $\mathbb{R}^3$ with standard orientation, I'm asked to expres the orientations of the six faces of the cube as differential 1-forms evaluted at the corss product of $v_{1}$ and $...
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1answer
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Multivariable Calculus, Help with pre-Bachelos Homework [closed]
I need help with a couple of problems for homework, btw I need to finish this in less than 12 hours, I'll appreciate your help, and srry if something is wrong or is confusing I'm not native speaker.
...
