Questions tagged [stokes-theorem]
Stokes' theorem is a result about integration of differential forms.
0
votes
1answer
67 views
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 $\Gamma:=\partial \Omega$, $...
4
votes
1answer
84 views
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 ...
0
votes
2answers
38 views
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 ...
0
votes
0answers
18 views
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 ...
0
votes
0answers
20 views
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!
3
votes
1answer
46 views
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 ...
0
votes
0answers
28 views
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 ...
0
votes
0answers
13 views
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 ...
1
vote
0answers
87 views
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 ...
0
votes
0answers
14 views
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 $\...
0
votes
0answers
44 views
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 \...
1
vote
0answers
19 views
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 ...
1
vote
0answers
35 views
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 ...
0
votes
0answers
41 views
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. ...
2
votes
0answers
56 views
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 ...
0
votes
2answers
59 views
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$, ...
1
vote
1answer
34 views
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$.
...
0
votes
1answer
23 views
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 ...
0
votes
0answers
35 views
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).$$
...
0
votes
0answers
27 views
Existance of $\phi \in L^2$ such as $L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$
[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$.
And ...
0
votes
0answers
13 views
Verifying Stokes' theorem with a disc
$ \iint_S \nabla \times \mathbf F \cdot dS = \int_{\partial S} \mathbf F \cdot dr $
Where S is the disc $ x^2 + y^2 \le 1$ in the plane $z = 0 $
The vector Field is $ \mathbf F = x^2i + (2xy+x)j + ...
1
vote
0answers
15 views
Looking for collection of exercises on Greene's theorem, Stokes theorem and the Divergence theorem
As the title states, I am looking for resources containging excersises on Greene's theorem, Stoke's theorem and the Divergence theorem.
Ideally the excersises would be of computational nature (i.e. ...
0
votes
3answers
56 views
Parameterizing the boundary curve of the surface defined by $x+y+z \geq 1$ and $x^2 +y^2+z^2=1$
I am unsure how to parameterize the boundary curve of the surface defined by
$x+y+z \geq 1$ and $x^2 +y^2+z^2=1$,
where $x,y$ and $z$ are real numbers.
The boundary curve should be the circle ...
1
vote
0answers
38 views
Prove Green Theorem using Gauss Theorem
Problem. Consider the General Stokes Theorem and $M$ a submanifold of $\mathbb{R}^{n}$, with boundary orientable.
(a) Prove the Green Theorem
$$\int_{M}(g_{x} - f_{y})dxdy = \int_{\partial M} ...
1
vote
0answers
21 views
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. $S=\{(x,y,z) / x\geq 0, y \geq 0, ...
2
votes
0answers
69 views
Stokes' theorem proof (is it acceptable?)
Is this an acceptable proof for Stokes' theorem? I'm currently creating my own Multivariable Calculus course and this is what I have for the proof.$\newcommand{\curl}{\operatorname{curl}}$
Also, I ...
1
vote
0answers
28 views
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 $\;$ ...
0
votes
0answers
23 views
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 ...
0
votes
0answers
25 views
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 ...
0
votes
0answers
14 views
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 $...
3
votes
1answer
61 views
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{...
0
votes
1answer
38 views
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 ...
8
votes
1answer
109 views
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 ...
2
votes
1answer
158 views
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 ...
0
votes
1answer
44 views
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\...
2
votes
1answer
30 views
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 $\int_M curl(f)\cdot n dS$ ...
2
votes
0answers
43 views
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 $...
1
vote
1answer
54 views
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 ...
0
votes
1answer
36 views
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 ...
0
votes
0answers
27 views
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 $Gamma$:
$$\...
1
vote
0answers
47 views
An intuitive explanation for Green theorem and Divergence theorem
As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize.
I think I have found a pretty intuitive way to think about the ...
1
vote
1answer
46 views
Prove that the integral of a differential form is zero
Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^n$ such that $[\mathbb{X},\mathbb{Y}]=0$, and let $\Phi_t$ and $\Psi_s$ denote their respective flows. Let $c:[0,1]\times [0,1]\to \...
1
vote
2answers
82 views
Application of Stokes' Theorem with the vector field $K=(-zy,zx,z^2)$ and the surface $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$
I'm asked to evaluate the flux $$\int_{S} rot K d\omega,$$ where $S$ is the region $-2 \leq z \leq 1, \sqrt{x^2+y^2}=1+z^2$ and $K$ is the vector field $(-zy,zx,z^2)$.
So when I evaluate directly the ...
7
votes
2answers
139 views
Apparent paradox when we use the Kelvin–Stokes theorem and there is a time dependency
I am having trouble to understand what is going on with the Maxwell–Faraday equation:
$$\nabla \times E = - \frac{\partial B}{\partial t},$$
where $E$ is the electric firld and $B$ the magnetic field. ...
2
votes
1answer
59 views
What goes wrong with Stokes theorem if a surface is not orientable?
For the Möbius strip parametrized by $\{\sigma(\theta,r)=((1+r\sin(\theta/2))\cos\theta,(1+r\sin(\theta/2))\sin\theta,r\cos(\theta/2))\ \mid \\ (\theta,r)\in A=(0,2\pi)\times(-1/2,1/2) \}$
we get ...
-1
votes
1answer
206 views
Verify Stokes' theorem on a hemisphere using a line integral
When solving this question, I wrote $r(t)$ as $\langle\cos t,0,\sin t\rangle$ then I differentiated it and got $dr=\langle-\sin t,0,\cos t\rangle\,dt$, then i substituted in the formula for line ...
0
votes
1answer
30 views
Determining $R$ so that the flux of the F field through S is maximum
For any $R>0$ let the surface $S={(x,y,z):x^2+y^2+(z-1)^2 = R^2; z≥1}$ oriented with the normal whose third coordinate is positive. And let the field $F(x,y,z) = (zx-x cos (z), -zy+ y cos(z), 4-x^...
6
votes
1answer
98 views
Where is my mistake? Calculating surface integral/Stoke's theorem
Let
$F(x,y,z)= \begin{pmatrix} -y \\ 2x\\z \end{pmatrix}$
be a vector field and $A$ a hemisphere
with $x^2+y^2+z^2=9 $ , $ z>0 $
with a circular edge at the $x,y $- level with the unit normal ...
0
votes
1answer
85 views
Volume form on a compact manifold is not exact
I am trying to show that a volume form $\mu$ on a compact manifold $M$ is not exact, i.e. show there is no $\alpha \in \Omega^{n-1}(M)$ such that $d\alpha = \mu$.
My attempt is the following: Suppose,...
1
vote
0answers
68 views
Intuition on Stokes' Theorem
Stokes' Theorem says that $$\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 ...
