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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questions
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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
$$ \oint f \vec{\bigtriangledown}g \cdot d\vec{l} = \int_S ((\vec{\bigtriangledown}f)\times(\vec{\...
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2answers
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How do I calculate the area in this problem?
I'm trying to solve this exercise about Stokes Theorem:
prove the validity of Stokes Theorem when $F(x,y,z)=(y,2x,2z)$ and $S$ is the part of the plane $z=y$ that lies in the cylinder $x^2+y^2=4y$ and ...
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Calculating stokes theorem using surface integrals for a surface comprising of 2 different surface
Image of question
Just a bit lost in approaching b ii), I tried doing this with 2 surface integrals, parametrized in cylindrical coords, one for each section but it seemed a bit lost as I continued ...
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Two surfaces in Stokes theorem [closed]
A surface S that is made of S1 and S2, S1 is a cilinder and S2 a semi-sphere, it's a clinder with a semi-sphere on top
$$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 ...
2
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0answers
81 views
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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1answer
28 views
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: $$...
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0answers
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Applications of the Generalized Stokes Theorem?
It states that for a $k$-dimensional manifold (with boundary) $M$ and $k-1$-differential form $f$ we have
$$\int_{\partial M}f = \int_M df.$$
Most of textbooks spend significant effort to prove the ...
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1answer
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Stokes in physics context: magnetic moment
I've got a problem where I come a little unprepared not knowing how to tackle:
Given a Vector field $$\textbf{W}: \mathbb{R^3} \rightarrow\mathbb{R^3}, \quad \textbf{W(x)} = \vec{b} \times \vec{x}$$
...
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1answer
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Rudin “Principles of Mathematical Analysis” 10.31 Positively oriented boundaries
I wonder if someone can explicitly construct a proof for the following statement mentioned in Rudin chapter 10 differential form:
If $T_1, T_2$ are injective mappings of the n-simplex $Q^n$ into $\...
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1answer
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Use Stokes Theorem to Prove Two Integrals on Differentiable Manifiolds are equivalent
I have to answer the following problem:
Let $\omega= ydx + xzdy + xdz$.
Let $S_1$ be the portion of the upper hemisphere given by $\phi_1(u,v)=(u,v,\sqrt{4-u^2-v^2})$; where $u^2 + v^2 \leq 2$.
Let $...
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0answers
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Unit normal Vector or normal normal vector for flux?
In my script flux through a compact manifold $M$ with boundary $\partial M$ is defined by: $$\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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1answer
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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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3answers
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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 ...
2
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1answer
31 views
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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0answers
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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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2answers
55 views
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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2answers
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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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0answers
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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
$$
\begin{align}
& \iint_{\partial E} F \cdot d S \\[6pt]
\text {where } & F=\left(x y^2+...
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3answers
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Evaluate the line integral $\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$ using Stokes' Theorem
I found the following problem in a textbook (translated):
Evaluate $$\int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz$$ where $\;\gamma\;$ is given by the intersection ...
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1answer
58 views
Evaluate the surface integral $\iint_\Sigma (y^2-z^2)e^{yz}\,ds$ using Stokes' Theorem
I found the following problem in a textbook:
Evaluate $$\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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1answer
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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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0answers
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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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1answer
54 views
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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2answers
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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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1answer
31 views
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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0answers
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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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0answers
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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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1answer
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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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0answers
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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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1answer
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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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0answers
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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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1answer
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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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0answers
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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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votes
1answer
38 views
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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1answer
43 views
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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1answer
148 views
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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1answer
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Calculate and find the flux integral
Find the flux integral $$\iint_S \operatorname{rot} \vec{F} {N}\,dS$$ where $S$ is the half sphere $x^2+y^2+z^2=4$ and $z \ge 0$ with an aligned unit standard $\vec{N}$ (normal) and $\vec{F} =(3x-y, ...
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1answer
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$\iint_A\nabla\times\textbf{u}\cdot \textbf{n}\ dS$ with $\nabla\times\textbf{u}$ known.
Let $$\textbf{v}=\operatorname{curl}\textbf{u}= (-2+43x-86y+43z, 3-2x+4y-2z, 2-47x+94y-47z)$$ and let $A$ be the part of $z(x,y)=9-x^2-9y^2$ that is above the plane $z(x,y)=x+45y+\frac{113}{2}$. Now ...
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0answers
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Using Stokes theorem on a discontinuous function
I've a vector function whose curl is well defined inside and outside,except on the boundary
$\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$
I'm interested in using stokes theorem across the ...
2
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1answer
70 views
Prove that if $f:M\rightarrow\Bbb R$ is a scalar function over a 1-manifold M without boundary then $\int_M df=0$
Well James Munkres in the text Analysis on Manifolds prove the general Stoke's theorem for $k$-form when $k>1$ and then he proves it for $k=1$ only when the bounary of the Manifold is not empty and ...
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1answer
33 views
Verifying Stokes’ Thm. $\vec F = \left< -y,x,-2\right>$ where $S$ is the region defined by $z^2 = x^2 + y^2, \space z \in [0,4]$
I want to verify Stokes’ Theorem by evaluating both the left and right side of the following eqn.
$$\int_C \vec F \cdot d\vec r = \int \int_S curl \vec F \cdot d\vec S$$
where $\vec F = \left< -y,x,...
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1answer
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Stoke's Thm. $\vec F = \langle x^2y,\frac{x^3}{3},xy \rangle $ and $C$ is the boundary of the intersection of $x^2+y^2=1$ and $z = y^2-x^2$
I am having trouble setting up the limit of integration for the following problem.
$\vec F = \langle x^2y,\frac{x^3}{3},xy \rangle $ and $C$ is the boundary of the intersection of $x^2+y^2=1$ and $z = ...
0
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2answers
41 views
Stokes' Thm. $\vec F = <2y,xz,x+y>$ through intersection of $x^2+y^2=1$ and $z=y+2$
I am trying to evaluate $\int_C \vec F \cdot d \vec r$ where
$\vec F = <2y,xz,x+y>$ and $C$ is the intersection of $x^2+y^2=1$ and $z=y+2$.
I believe that $$curl \vec F = <1-x, -1, z-2> $$
...
1
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1answer
29 views
Can I get some help with a line integral? $\int_\Gamma (x^2 + 5y + 7z)dx + (y^2+z+5x)dy + (z^2+7x + y)dz)$
A(0,10, 0), B($3 \sqrt 10, 0, \sqrt10$), C(2, 8, 0) with $\int_\Gamma (x^2 + 5y + 7z)dx + (y^2+z+5x)dy + (z^2+7x + y)dz)$
$\Gamma$ is made from the following segments:
A->B {$ (x,y,z)∈ \Bbb R^3 / ...
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votes
2answers
27 views
Stoke's Thm. $\vec F = (x+y^2, y+z^2, z+x^2)$ and $S$ is the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$
I am trying to evaluate
$$\int _C \vec F \cdot d\vec r$$
using Stoke's Thm. where $\vec F = (x+y^2, y+z^2, z+x^2)$ and $S$ is the triangle with vertices $(1,0,0), (0,1,0), (0,0,1)$
So, I can see why ...
0
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0answers
27 views
One-point gradient estimator and Stokes' theorem
In bandit convex optimization, we are only given access to zeroth-order oracle of a function but not first-order (gradient) oracle. Hence, people often use some one-point gradient estimator to ...
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0answers
24 views
Stokes theorem to prove that a k-form is closed
I'm trying to use Stokes theorem to prove the following: if $\omega$ is a $C^1$ k-form on $\mathbb{R}^n$ such that $\int_M \omega =0 $ for any compact oriented manifold M of dimension k, then $\omega$...
3
votes
1answer
67 views
compute $\int_\gamma F\cdot dr$ using Stokes
Evaluate
$$ \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 $xy$ ...
