Questions tagged [stokes-theorem]

For questions about Stokes' theorem. Stokes' theorem relates the integral of a differential form over the boundary of some orientable manifold M is equal to the integral of its exterior derivative over the whole of M.

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Line integral relations

Hi I was consulting this forum and I found this question where the user says he proves these relations $$ \oint f \vec{\bigtriangledown}g \cdot d\vec{l} = \int_S ((\vec{\bigtriangledown}f)\times(\vec{\...
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How do I calculate the area in this problem?

I'm trying to solve this exercise about Stokes Theorem: prove the validity of Stokes Theorem when $F(x,y,z)=(y,2x,2z)$ and $S$ is the part of the plane $z=y$ that lies in the cylinder $x^2+y^2=4y$ and ...
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Calculating stokes theorem using surface integrals for a surface comprising of 2 different surface

Image of question Just a bit lost in approaching b ii), I tried doing this with 2 surface integrals, parametrized in cylindrical coords, one for each section but it seemed a bit lost as I continued ...
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Two surfaces in Stokes theorem [closed]

A surface S that is made of S1 and S2, S1 is a cilinder and S2 a semi-sphere, it's a clinder with a semi-sphere on top $$S1: x^2+y^2=1, 0\leq z\leq1$$ $$S2: x^2+y^2+(z-1)^2=1, z\geq 1$$ $$\vec F=(xz+...
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Use Stokes' theorem to find area of a surface

Can you think of a way to apply Stokes' theorem to find the area of a surface in $\mathbb{R^3}$ by constructing some vector field $F(x, y, z)$ and taking its line integral over the boundary of the ...
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Interpretation of Integral of Differential 3-Form over 3-Manifold in Engineering?

So, the standard interpretation of a differential 1-form's integral over a compact, connected, oriented 1-manifold is work, and the standard interpretation of a differential 2-form's integral over a ...
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Verify Stokes theorem: semi-circle in $\mathbb{R}^2$

I'd like to verify stokes theorem for the Manifold given by $$M = \{(x,y)\in\mathbb{R^2}\vert \: x^2+y^2<25, x\geq0\} \cup\{(x,y)\in\mathbb{R^2}\vert \: x = 0, -5<y<5\}$$ in a vector field: $$...
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Applications of the Generalized Stokes Theorem?

It states that for a $k$-dimensional manifold (with boundary) $M$ and $k-1$-differential form $f$ we have $$\int_{\partial M}f = \int_M df.$$ Most of textbooks spend significant effort to prove the ...
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Stokes in physics context: magnetic moment

I've got a problem where I come a little unprepared not knowing how to tackle: Given a Vector field $$\textbf{W}: \mathbb{R^3} \rightarrow\mathbb{R^3}, \quad \textbf{W(x)} = \vec{b} \times \vec{x}$$ ...
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Rudin “Principles of Mathematical Analysis” 10.31 Positively oriented boundaries

I wonder if someone can explicitly construct a proof for the following statement mentioned in Rudin chapter 10 differential form: If $T_1, T_2$ are injective mappings of the n-simplex $Q^n$ into $\...
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Use Stokes Theorem to Prove Two Integrals on Differentiable Manifiolds are equivalent

I have to answer the following problem: Let $\omega= ydx + xzdy + xdz$. Let $S_1$ be the portion of the upper hemisphere given by $\phi_1(u,v)=(u,v,\sqrt{4-u^2-v^2})$; where $u^2 + v^2 \leq 2$. Let $...
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Unit normal Vector or normal normal vector for flux?

In my script flux through a compact manifold $M$ with boundary $\partial M$ is defined by: $$\Phi_M(F) = \int_M\langle F,N\rangle\,\mathrm{dA}$$ where $N$ is the unit normal vector. However I did see ...
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Flux through a plane with proper parametrisation

I've given the plane $$a+x\,b+y\,c$$ where $a = (1,0,0), b = (0,1,0), c = (0,0,1)$, $x,y \in[0,1]$ plus a vector field $F = (x\,y, y\,z, x\,z)$ now flux is defined by $\Phi(F) = \int\langle F,N\...
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Having difficulties to prove this relation (Stokes Theorem)

Let the field, $\vec{F}$ be class $C¹$ in the open $Ω$. And $\sigma_1$ and $\sigma_2$ are portions of regular surfaces with boundaries $\Gamma_1$ and $\Gamma_2$ oriented positively in relation to the ...
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Using Stokes theorem to find the line integral over the boundary of a paraboloid in the first octant opening downward the z-axis

I've been trying at this problem on my homework, but I think I am going about it the wrong way. I tried breaking it down into the line integrals of the boundaries of the surface, but I think I might ...
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Stokes' Theorem - cylindrical coordinates

I'm currently having an issue with verifying the validity of Stokes' Theorem on a particular problem. I can solve the problem by using Stokes' theorem to turn a surface integral of the curl of a ...
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Rudin Stokes' Theorem 10.33

I am wondering if someone could explain this last step that Rudin makes to conclude the proof. Confusion is highlighted Earlier parts he references With the integral over $dx_r$, I expanded $dx_r$ as ...
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How to choose the right bounds for this Stokes' theorem problem

There is a part of a cylinder $y^2 + z^2 = 1$ between $x =0$ and $x = 3$ and above the xy-plane. The boundary of this surface is a curve C which is oriented counter-clockwise when viewed from above. ...
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Multivariable divergence theorem work

Find the flux of the vector field $\vec{G}=\operatorname{curl} \vec{F}$, where $$ \vec{F}(x, y, z)=y^{3} \vec{i}+x^{3} \vec{j}+z^{3} \vec{k} $$ through the upward oriented part of paraboloid $$ z=1-x^{...
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Multivariable divergence theorem two cylinder question [duplicate]

Let $E$ be the intersection of the cylinders $x^{2}+y^{2} \leq 1, y^{2}+z^{2} \leq 1$. Compute the flux $$ \begin{align} & \iint_{\partial E} F \cdot d S \\[6pt] \text {where } & F=\left(x y^2+...
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Evaluate the line integral $\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$ using Stokes' Theorem

I found the following problem in a textbook (translated): Evaluate $$\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$$ where $\;\gamma\;$ is given by the intersection ...
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Evaluate the surface integral $\iint_\Sigma (y^2-z^2)e^{yz}\,ds$ using Stokes' Theorem

I found the following problem in a textbook: Evaluate $$\iint_\Sigma (y^2-z^2)e^{yz}\,ds\,,$$ where $\;\Sigma\;$ is given by $x^2+y^2+z^2=1$, $\;z\geq0$, by evaluating the rotation of the field $\;\...
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How to choose area of integration for Stokes' theorem on a submanifold of a torus?

This question is coming from physics, so I hope you will be patient as you read the following. Setup: Assume all criteria for applying Stokes' theorem are met. Consider a 2D plane representation of a ...
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How to deal with boundary orientation in Exercise 16-5 from Lee's Introduction to Smooth Manifolds

16-5. Suppose $M$ and $N$ are oriented, compact, connected, smooth manifolds, and $F,G:M\to N$ are homotopic diffeomorphisms. Show that $F$ and $G$ are either both orientation-preserving or both ...
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Stokes' theorem applied to a cylinder constraint to the first quadrant

I am helping a friend to solve a calculus list. I have solved all the questions, except, this one. Should you know how to solve, please, let me know! Solve $\int_C \vec F \cdot dr$ by Stokes theorem, ...
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Showing that $\oint_C(y, z, x)\cdot \mathbf{dr} = \sqrt{3}\pi a^2$ with the use of Stokes' theorem

I have to show that $\oint_C(y, z, x)\cdot \mathbf{dr} = \sqrt{3}\pi a^2$ when $C$ is the curve of intersection of $x^2 + y^2 + z^2 = a^2$ and $x + y + z = 0$. I currently confused as at the end I ...
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Closed line integral with spherical coordinates

So I had this question in my homework, but I could not solve it. The question is as follows: There is a closed path going from (0,0,0) to (1,0,0) to (0,1,0) to (0,1,2) to (0,0,0). I have first ...
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Using Green's Theorm to find area pictorially

This is a follow-on to Dr. Shifrin's comment about depicting the derivative of a polar coordinate transformation. He said "The row vectors you don't see so much geometrically." I agree ...
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Clarification on the surface integral in Stokes' theorem

I am looking to understand what exactly is calculated in the surface integral of Stoke's theorem. As a context, consider the exercise 1. of section 16.5. in 9th edition of Calculus: "Evaluate $\...
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Conversion of formula about Stokes' theorem

$\int \nabla\times\vec{F}\cdot{\hat{n}}ds=\iint(-\frac{\partial z}{\partial x}(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})-\frac{\partial z}{\partial y}(\frac{\partial P}{\partial z}-\...
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Stokes' theorem on a portion of a sphere

I am trying to evaluate the surface integral $\int\int_{\partial S}(\nabla \times F)\cdot n dA$ on the surface $S = \{(x, y, z)\mid 0 \leq z \leq 1, x^2 + y^2 + (z - 1)^2 \leq 4\}$. The hint I have ...
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Does Stokes law on a non-conservative field?

I was trying to check if Stokes would hold in a field $(3x^2y^3,-x^3y^2)$ (non conservative as curl is non zero) for the figure below but I got that LHS (the surface integral) was not equal to RHS (...
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Can a field be a gradient of a scalar?

I am trying to answer a question in my electromagnetics book (Cheng). I am given a field A with non-zero curl. It is then asked if A can be expressed as a gradient of a scalar. Note that its not ...
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Finding the integral limits of a parametrized triangle for Stokes' theorem

I suppose my question is really how to parametrize points of the triangle. But for context, suppose the three vertices of a triangle are $(-5, 1, 0), (0, -5, 1), (1, 0, -5)$, the coordinate functions ...
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Functional derivative and non-abelian Stokes theorem

I have the following expression to calculate: \begin{equation*} \frac{\delta}{\delta \omega^\mu_{ab}}F\left[ \int_{\partial P} \omega\right] \end{equation*} Where $\omega$ is a vector-valued $1$-form ...
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Is $\nabla \cdot (\nabla\times \textbf{F})$ = div $\textbf{F} $? [duplicate]

Is $\nabla \cdot (\nabla\times \textbf{F})$ = div $\textbf{F} $? Where $(\nabla\times \textbf{F})$ = rot $\textbf{F}$. If yes, why?
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Vectorial $L^p$ spaces

Let $\Omega \subset \mathbb{R^n}$ Denote: $Ł^p(\Omega):=(L^p(\Omega))^N=L^p(\Omega)×L^p(\Omega)×...×L^p(\Omega)$, $N$ times, as the vectorial $L^p$ space associated to the scalar one, where $N \ge 1$. ...
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Stokes theorem for a current

I am struggling to understand the Stokes theorem for currents(differential form with distributional coefficients). The statement is as follows: Lest $S$ be a current of degree $N - 1$ with compact ...
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Calculate and find the flux integral

Find the flux integral $$\iint_S \operatorname{rot} \vec{F} {N}\,dS$$ where $S$ is the half sphere $x^2+y^2+z^2=4$ and $z \ge 0$ with an aligned unit standard $\vec{N}$ (normal) and $\vec{F} =(3x-y, ...
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$\iint_A\nabla\times\textbf{u}\cdot \textbf{n}\ dS$ with $\nabla\times\textbf{u}$ known.

Let $$\textbf{v}=\operatorname{curl}\textbf{u}= (-2+43x-86y+43z, 3-2x+4y-2z, 2-47x+94y-47z)$$ and let $A$ be the part of $z(x,y)=9-x^2-9y^2$ that is above the plane $z(x,y)=x+45y+\frac{113}{2}$. Now ...
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Using Stokes theorem on a discontinuous function

I've a vector function whose curl is well defined inside and outside,except on the boundary $\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$ I'm interested in using stokes theorem across the ...
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Prove that if $f:M\rightarrow\Bbb R$ is a scalar function over a 1-manifold M without boundary then $\int_M df=0$

Well James Munkres in the text Analysis on Manifolds prove the general Stoke's theorem for $k$-form when $k>1$ and then he proves it for $k=1$ only when the bounary of the Manifold is not empty and ...
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Verifying Stokes’ Thm. $\vec F = \left< -y,x,-2\right>$ where $S$ is the region defined by $z^2 = x^2 + y^2, \space z \in [0,4]$

I want to verify Stokes’ Theorem by evaluating both the left and right side of the following eqn. $$\int_C \vec F \cdot d\vec r = \int \int_S curl \vec F \cdot d\vec S$$ where $\vec F = \left< -y,x,...
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Stoke's Thm. $\vec F = \langle x^2y,\frac{x^3}{3},xy \rangle $ and $C$ is the boundary of the intersection of $x^2+y^2=1$ and $z = y^2-x^2$

I am having trouble setting up the limit of integration for the following problem. $\vec F = \langle x^2y,\frac{x^3}{3},xy \rangle $ and $C$ is the boundary of the intersection of $x^2+y^2=1$ and $z = ...
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Stokes' Thm. $\vec F = <2y,xz,x+y>$ through intersection of $x^2+y^2=1$ and $z=y+2$

I am trying to evaluate $\int_C \vec F \cdot d \vec r$ where $\vec F = <2y,xz,x+y>$ and $C$ is the intersection of $x^2+y^2=1$ and $z=y+2$. I believe that $$curl \vec F = <1-x, -1, z-2> $$ ...
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Can I get some help with a line integral? $\int_\Gamma (x^2 + 5y + 7z)dx + (y^2+z+5x)dy + (z^2+7x + y)dz)$

A(0,10, 0), B($3 \sqrt 10, 0, \sqrt10$), C(2, 8, 0) with $\int_\Gamma (x^2 + 5y + 7z)dx + (y^2+z+5x)dy + (z^2+7x + y)dz)$ $\Gamma$ is made from the following segments: A->B {$ (x,y,z)∈ \Bbb R^3 / ...
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Stoke's Thm. $\vec F = (x+y^2, y+z^2, z+x^2)$ and $S$ is the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$

I am trying to evaluate $$\int _C \vec F \cdot d\vec r$$ using Stoke's Thm. where $\vec F = (x+y^2, y+z^2, z+x^2)$ and $S$ is the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$ So, I can see why ...
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One-point gradient estimator and Stokes' theorem

In bandit convex optimization, we are only given access to zeroth-order oracle of a function but not first-order (gradient) oracle. Hence, people often use some one-point gradient estimator to ...
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Stokes theorem to prove that a k-form is closed

I'm trying to use Stokes theorem to prove the following: if $\omega$ is a $C^1$ k-form on $\mathbb{R}^n$ such that $\int_M \omega =0 $ for any compact oriented manifold M of dimension k, then $\omega$...
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compute $\int_\gamma F\cdot dr$ using Stokes

Evaluate $$ \int_\gamma F\cdot dr$$ where $ F=(ye^x,e^x+x^3,z^5)$ and $ \gamma\ $ is the intersection between $x^2+y^2=1 $ and $z=2xy$ oriented in such a way that the orthogonal projection on the $xy$ ...

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