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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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' ...
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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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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. ...