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Questions tagged [stokes-theorem]

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

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What is the result of the integration by parts of $\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega$?

$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded domain with Lipschitz continuous and piecewise smooth boundary $\Gamma:=\partial \Omega$, $...
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Is $\int_X d(F^{*}\omega) = 0$ because of a corollary of Stokes' Theorem?

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. The last part of Proposition 11.11 goes $$\int_X d(F^{*}\omega) = \int_X F^{*}(d\omega) = 0$$ Can we just skip commutativity ...
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How can I parameterize the intersection of $x+y+z=0$ and $x^2 +y^2 + z^2 =a^2$?

I'm doing exercises using stokes theorem. And i'm looking for parameterization of the intersection: $x+y+z=0$ and $x^2 +y^2 + z^2 =a^2$ in terms of x and y such that $r(x,y)$ . This is because ...
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Clarification of how to use Stokes' Theorem if you are not given a Vector Field

I am working on a few surface integrals in preparation for an exam and one question specifically states to use Stokes' Theorem to solve, however, rather than giving a vector field, we are given a ...
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Stokes Theorem for circulation around triangle

I know that in order to find this I have to use Stoke's Theorem, but I don't know how to do that. Any help would be great, thank you in advance!
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Is it “Valid” to prove Stokes' Theorem with Green's Theorem?

In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in ...
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How to convert spherical co-ordinates of a vector field to cartesian co-ordinates :

v$(r,φ,θ) = (r cos2 θ)$r$ − (rcosθsinθ)$θ$ + 3r$φ, where r, θ and φ are the unit spherical vectors. I was trying to calculate the line integral of the function along the path described in the ...
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How to get the divergence theorem to work in this case?

Let $\displaystyle\mathbf{v}_2=\frac{1}{r^2}\hat{\mathbf{r}}$. I found that $\displaystyle\nabla.\mathbf{v}_2=0$ everywhere except at the origin, where it is not defined. So, we cannot use the ...
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How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
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What is a simple parametric surface?

What is the formal definition of simple parametric surface? We are dealing in $\mathbb R^3$. I came across the term in Apostol's Calculus (2nd volume), where he states that the parametric function $\...
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Stokes theorem in cylindrical coordinates

Consider the following vector field in cylindrical polar components $$\mathbf{F} = \frac{z}{\rho }(1-e^{-\rho /\gamma} ) \hat{\mathbf{e}}_{\rho } +\frac{z}{\rho } \hat{\mathbf{e}}_{\phi } + \phi \...
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Generalized Stokes Theorem, applied to 2D/3D

Generalized Stokes Theorem says $\int_{\partial S} \omega = \int_S d \omega$ where $\omega$ is a $(k-1)$-form and $S$ is a $k$-dimensional manifold. I understand that, for $n=3$, If $k=3$ this ...
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LHS and RHS of Stokes' theorem not equal

Question: I am trying to test my understanding of Stokes' theorem by calculating the left and right hand side of the theorems equality by way of example and seeing whether they equal each other. They ...
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How to evaluate $\iint (\nabla \times \vec F) \cdot d \vec a$

C is the curve arising from the intersection of the elliptic cylinder $\frac{(x-2)^2}{36} + \frac{(y-1)^2}{9} = 1$ and the plane $3x + y + z = 1$, which circulates counterclockwise as seen from above. ...
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Stokes theorem and Volume forms

I have the following short argument which seems to say that there are no non-vanishing, (n-1)-forms on a closed manifold of dimension n but I am not very confident in my understanding of a "volume ...
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Evaluating $\oint_C \vec{F} \cdot d\vec{r}$

After trying a couple of times, but failing to find a way to solve these problems, I decided I should perhaps ask the people on this forum for help. Problem 1 Let $C$ be the curve $(x-1)^2+y^2=16$, ...
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Finding the maximum value of $\oint_C \vec{F} \cdot d\vec{r}$ of a vector field

I'm stuck with trying to find the maximum value for $\oint_C \vec{F} \cdot d\vec{r}$ of the vector field $\vec{F}=\langle5z, x, y \rangle$ where $C$ is a simple closed curve in the plane $2x+3y+z=7$. ...
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Value of exterior derivative as of limit of integral

Let $w$ be a $(n-1)$-form in $\mathbb{R}^n$, and $e_1,...,e_n$ be a basis for the tangent space at $x\in\mathbb{R}^n$. Let $B(x,r)$ be the ball centered at $x$ with radius $r$ and $S(x,r)$ it's ...
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Calculate the flux integral (on unit sphere).

Consider a matriz of rotation in $z$-axis $$R_{\theta} = \left(\begin{array}{ccc} \cos\theta & -\sin \theta & 0 \\ \sin\theta & \cos\theta& 0 \\ 0 &0&1\end{array}\right).$$ ...
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Existance of $\phi \in L^2$ such as $L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$

[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere] Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$. And ...
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Verifying Stokes' theorem with a disc

$ \iint_S \nabla \times \mathbf F \cdot dS = \int_{\partial S} \mathbf F \cdot dr $ Where S is the disc $ x^2 + y^2 \le 1$ in the plane $z = 0 $ The vector Field is $ \mathbf F = x^2i + (2xy+x)j + ...
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Looking for collection of exercises on Greene's theorem, Stokes theorem and the Divergence theorem

As the title states, I am looking for resources containging excersises on Greene's theorem, Stoke's theorem and the Divergence theorem. Ideally the excersises would be of computational nature (i.e. ...
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Parameterizing the boundary curve of the surface defined by $x+y+z \geq 1$ and $x^2 +y^2+z^2=1$

I am unsure how to parameterize the boundary curve of the surface defined by $x+y+z \geq 1$ and $x^2 +y^2+z^2=1$, where $x,y$ and $z$ are real numbers. The boundary curve should be the circle ...
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Prove Green Theorem using Gauss Theorem

Problem. Consider the General Stokes Theorem and $M$ a submanifold of $\mathbb{R}^{n}$, with boundary orientable. (a) Prove the Green Theorem $$\int_{M}(g_{x} - f_{y})dxdy = \int_{\partial M} ...
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Calculate line integral in a vector field

I have the following problem and I'm not able to solve it. Given $F(x,y,z) = (1-2z, 0, 2y)$, calculate the line integral of C, where C is the contour of the surface. $S=\{(x,y,z) / x\geq 0, y \geq 0, ...
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Stokes' theorem proof (is it acceptable?)

Is this an acceptable proof for Stokes' theorem? I'm currently creating my own Multivariable Calculus course and this is what I have for the proof.$\newcommand{\curl}{\operatorname{curl}}$ Also, I ...
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What does the right hand side of this Stokes' theorem mean?

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." I don't understand what the right hand side of the formula in the following theorem means. 4.9 Stokes' Theorem II $\;$ ...
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Reference request on an equation while studying Stokes steam-functions

I came across this equation while studying Stokes steam-functions and I'm not sure how its derived. The equation is $$(r^2-\frac{a^3}{r})\sin^2(\theta)=b^2$$ I believe it is the equation of a ...
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how to calculate $\int_C~ydx + zdy + xdz$, where $C$ is oriented counterclockwise, as viewed from above. ??

Let $C$ be the boundary of the portion of ${z = x ^2 + y ^2}$ below the plane {$6x + 2y + z = 20$}. Find $\int_C~ydx + zdy + xdz$, where $C$ is oriented counterclockwise, as viewed from ...
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Show $\int_{dM} w=\int_M dw$, use $d(w|_M)$ not $w\in \Bbb A(\Bbb R^3)$ , $w|_M \in \Bbb A(M)$.

Let $w \in \Bbb A^1(\Bbb R^3)$, $w=xzdy-yzdx$, $M=\{z=f(x^2+y^2\} over $x^2+y^2 \leq \Bbb R^2$. Illustrate Stoke's theorem with $(M,w|M)$. For easier use $f(x)=x$. Show $\int_{dM} w=\int_M dw$, use $...
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Proving Fundamental Theorem of Calculus given Stokes' Theorem

Accepting Stokes' theorem (101) as a premise, how do I prove the fundamental theorem of calculus (102)? I know that the FTC is a straightforward specialization of Stokes' theorem and so proving $\text{...
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Prove Poincaré Lemma for $1$-form

Let $U\subseteq\mathbb{R}^n$ be an open set that contains $0$, and for all $t\in[0,1]$ and $ x\in U$, $tx\in\mathbb{R}^n$. Show that every closed differentiable 1-form $w$, (i.e. $dw=0$) is an exact ...
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Can the Gauss-Bonnet theorem be proven from Stokes's theorem?

In a comment to this question, John Ma claims that the Gauss-Bonnet theorem can be proven from Stokes's theorem, but does not explain how. For two dimensions, Stokes's theorem says that for any ...
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Question on the proof of Stokes' Theorem in Spivak

The following is a quick outline of the proof of Stokes' Theorem as found in a Comprehensive Introduction to Differential Geometry Vol. 1 by Spivak. Theorem (Local Stokes' Theorem). Let $M$ be a ...
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Interpretation of stokes theorem

I recently solved the following task: Let $A = [0,1]^3$ and $\omega = \dfrac{x_1^2 x_2^3}{1+x_3^2} \ dx_1 \wedge dx_3$ Show that this fulfills stokes theorem by showing that $\displaystyle \int_A \ d\...
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Calculating a flux integral using Stokes vs. directly

Let $G=\{(x,y,z)\in\mathbb R^3|x^2+y^2=1 , \quad 0\leq z\leq 1\}$ Let $f: \mathbb R^3\to\mathbb R^3,\quad f(x,y,z)=\begin{pmatrix}yz^2\\-x\\ye^z\end{pmatrix}$ Calculate $\int_M curl(f)\cdot n dS$ ...
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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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1answer
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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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1answer
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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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1answer
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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 ...