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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Stokes Theorem for the intersection of $z=x^2+y^2$ and $z=x+2$ and parameterizing the portion of the plane inside of $z=x^2+y^2$

I have been trying to solve the surface integral $$\int \int_S curl(\vec{F}) \cdot \vec{dS}$$ using stokes theorem for the vector field $$F(x,y,z)=\left(xz,yz,xy\right)$$ and where $S$ is the ...
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Confirmation of calculation: $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})dS$

The question goes like this: Evaluate $\displaystyle \iint_S (yz\,\hat{\imath}+zx\,\hat{\jmath}+xy\,\hat{k})\,dS$ where $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ in the first octant. I saw ...
Nεo Pλατo's user avatar
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Gradient and differential surface element

In my textbook There is the following step made when dealing with a certaion surface integral over a random surface: $$\int_S(\nabla\otimes\vec{v})\cdot d\vec{S}=\int_S\nabla(\vec{v}\cdot d\vec{S})$$ ...
Krum Kutsarov's user avatar
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Having trouble calculating $\int_C(y-2z)\text{d}x + (x-z) \text{d}y + (2x-y)\text{d}z$ using Stokes' theorem

$\text{Calculate using Stokes' theorem } I=\int_C(y-2z)\text{d}x + (x-z) \text{d}y + (2x-y)\text{d}z, \text{ where } C \text{ is the intersection between } x^2+y^2+z^2=a^2, a > 0, \text{ and } x - ...
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Differential surface element and nabla operator

If we have the vector field $\vec{u}=\vec{A}\times \vec{v}$, where $\vec{A}=\text{const.}$ and we integrate over some closed curve, by using Stokes' theorem we get: $$ \begin{align} \oint_{\partial S}...
Krum Kutsarov's user avatar
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Stokes Theorem and Metric Connection

I am looking at an expression like $$ \int_{\partial M}\langle D\psi, D\psi\rangle dx dy $$ where $\psi$ is some (well-behaved) function $\partial M \rightarrow \mathbb{C}$. $D=d+A$ is the connection ...
Makkabi's user avatar
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Stokes’ theorem manipulation using gauss divergence theorem

Apply Stokes theorem to prove that $\int_{c} ydx+zdy+xdz =-2\sqrt{2}\pi a^2$ Where C is the curve given by $x^2+y^2+z^2-2ax-2ay=0, x+y=2a$ ; and begins at the point (2a,0,0) and it goes first below ...
dreamboat's user avatar
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3 answers
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Evaluation of surface integral $\iint F ds$; $F(x,y,z)=(x,y,z)$

Let $F(x,y,z)=(x,y,z)$. $\\[10pt]$ Evaluate: $\iint_{S} F ds$; where $S$ is the upper hemisphere of radius $3$, centered at the origin. I defined $\phi(u,v)=(\sqrt{9-v^2}\cos(u),\sqrt{9-v^2}\sin(u),...
J P's user avatar
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Integral over torus of given $2-$form.

Let $D$ denote the torus of revolution in $\Bbb{R}^3$ by revolving the circle $$(y-2)^2+z^2=1,$$ along the $z-$axis. If $\omega = z \, dx \wedge dy$, compute $\int_D \omega$. Attempt: So I am ...
anonymous's user avatar
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Why Curl is Zero for the Magnetic Field of Infinite Wire

In cylidrical coordinates,the magnetic field around an infinite (thin) current-carrying wire is $\vec{B} =\frac{\mu_0I}{2 \pi r} \hat{\phi}$ Naiively computing the curl of this using Curl$\vec{B}$ = $\...
Harvey Williams's user avatar
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Calculating the flux over a non simply connected surface

Here is the question: The surface $S$ shown here has boundary the circle of radius $2$ in the $xz$-plane. With respect to the normal vector field indicated, compute the flux of $G = \langle 0, 3, 0 \...
Sudar Kartheepan's user avatar
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Stokes theorem on $C^1$ manifolds?

I have recently encountered Stokes theorem on embedded submanifolds of $\mathbb{R}^n$, and I didn't manage to find a proof for $C^1$ vector fields over $C^1$ manifolds, infact I have only seen that ...
Lorenzo Vanni's user avatar
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On elementary proof of Stokes' theorem

I have looked at several elementary proofs (i.e., using only basic calculus, and not using differential form or manifold) of Stokes' theorem in books and Wikipedia, and all seem to use the fact that ...
ashpool's user avatar
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Symplectic area is invariant under homotopy

Suppose $M$ is a compact, orientable surface with boundary, $N$ is a manifold with symplectic form $\omega$, and $F:I\times M\to N$ is a homotopy fixing the boundary $\partial M$. I'd like to show ...
subrosar's user avatar
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Getting negative value of Stokes theorem integral

I was attempting the question: Let $F(x,y,z) = (y, -x, 2z^2+x^2)$ and $S$ be the part of the sphere $x^2+y^2+z^2 = 25$ that lies below the plane $z = 4$. Evaluate the expression $\iint \operatorname{...
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What is the parametrization of this tetrahedron so that it satisfies assumption in Stokes' theorem?

This question is about a problem in Apostol's Calculus, Vol II, section 12.13 about Stokes' theorem. A question about this problem has been asked before, but that question is about solving the ...
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Need help confirming answer to this stokes theorem question

I'm trying to solve this question in preparation for my upcoming uni subject, i believe i understand it properly but im not quite sure, would like some assurance or guidance. (a) Use Stokes Theorem to ...
user1251062's user avatar
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Stokes / Green's theorem with non-regular regions

One statement of Green's Theorem (Stewart) I have seen is: Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$. If $\mathbf{F}...
Alex B's user avatar
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Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true

Im trying to learn aerodynamics in general for my course. Every video i see to derive the concept of greens and stokes theorem shows how the inner curls within a surface area cancel to 0 and its only ...
George kirby's user avatar
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When a surface patch of a non-closed surface is specified, does that automatically specify a set of vector fields due to Stokes' Theorem?

If I understand correctly, then any time we specify a particular surface patch of some non-closed surface, we also automatically specify not only the curve which bounds the surface patch but also the ...
Simon M's user avatar
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Stokes theorem not holding

I have a vector field $\vec{H} = (8z,0,-4x^3)$ Naturally, $\nabla \times \vec{H} = (0,8+12x^2,0)$ Stokes theorem says: $$ \int_s{\nabla \times \vec{H}} \cdot \vec{dS} = \oint_l{\vec{H} \cdot dl} $$ ...
rjpj1998's user avatar
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Stokes theorem 2 sides not matching with Magnetic waves

We have been asked to verify stokes theorem for a magnetic field. We know Stokes theorem states, for any vector field $\vec{H}$: $$\int_S{(\nabla \times \vec{H}) \cdot \vec{dS}} = \oint_L{\vec{H} \...
rjpj1998's user avatar
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Is there a generalization of the divergence theorem for integrals of the form $\iint_S \vec F \, dS$ [duplicate]

I know the divergence theorem can be applied to closed flux integrals $$\iint_S \vec F \cdot d\vec S = \iiint_V \nabla \cdot \vec F \, dV$$ but what if I have an integral of the form $$\iint_S F \, ...
user256872's user avatar
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Rewriting FTC to look like Stoke's Thm

The Fundamental Theorem of Calculus states $\int_{a}^{b} f' \ = f(b)-f(a)$ If I define $\frac{df}{dx}:=f'$ and $\int_{a}^{b} f \, dx \ := \int_{a}^{b} f \ $, then I can rewrite above as $\int_{a}^{b} \...
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Reference to statement: A harmonic function on a compact connected riemannian manifold is constant.

I read this statement a few times, for example in the answer to this question. Does anyone have a reference to this statement?
Butters's user avatar
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Converting a differential form to a measure

so today I was looking at the Generalized Stokes' Theorem: \begin{align} \intop_{\Omega} d\omega=\intop_{\partial\Omega}\omega\ \ , \end{align} where $\Omega$ is some region, and $\omega$ is a ...
Sora8DTL's user avatar
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Can i use Green's theorem to calculate the area of an abstract triangle on a plane?

I want to see some examples of Green's theorem used to calculate the area of some simple 2D shapes, but i haven't encountered a lot of them. My goal is to find (or study) a general procedure for ...
Simón Flavio Ibañez's user avatar
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Evaluating line integral with Stokes' Theorem

I have to use Stokes' Theorem to evaluate the line integral $$ \int_{\partial S} F \cdot dx $$ where $\partial S$ is the boundary of $$S =\{x^2 +y^2 = z^4,\, 0 \le z \le 3\}$$ and $$F = (xy, y, -2xz^2)...
Buhgro's user avatar
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Stokes' Theorem Always Surface Independent?

Is Stokes' Theorem always surface independent? In my textbook it says that if F has a vector potential A such that curl(A)=F, then the following is true: $$\iint F \cdot dS =\int A \cdot dr$$ Excuse ...
Dr-Galunga's user avatar
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GIven a surface area F, a normal vector, a point and a vector field, calculate the flow of the vector field through F.

the surface F, the vector field v, the point P and the normal vector n are defined as follows: $$ F = \vec{x}(s,t) = \begin{bmatrix}s \\t \\s^2+t^2 \end{bmatrix} \ s,t \in [0,1]$$ $$\vec{n} = \begin{...
giuli0110's user avatar
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Use Stokes' Theorem to evaluate this integral

I have this pretty simple exercise: Use Stokes' Theorem to find $\iint_S \text{curl} \, \mathbf{F} \cdot d \mathbf{S}$, where $\mathbf{F}(x,y,z) = ze^y \mathbf{i} + x \cos y \mathbf{j} + xz \sin y \...
iwjueph94rgytbhr's user avatar
2 votes
1 answer
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Stokes' Theorem and perimeter

As a consequence of Stokes' Theorem it seems that the perimeter of a closed curve $C$ can be obtained by choosing $F$ to be the vector field formed by rewriting the unit tangent $T$ of $C$ as a ...
Simon M's user avatar
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1 answer
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Apply Stokes theorem on a curve which is intersection of a sphere and plane

Apply Stokes theorem to prove that $\int_{c} ydx+zdy+xdz =-2\sqrt{2}\pi a^2$ Where C is the curve given by $x^2+y^2+z^2-2ax-2ay=0, x+y=2a$ ; and begins at the point (2a,0,0) and it goes first below ...
Sandeep's user avatar
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Nomenclature Stokes Theorem

I have a profound understanding of the classical Stokes theorem (aka Curl theorem). However, I am a little confused, how the theorem is written on Wikipedia $$ \iint_\Sigma (\nabla \times \mathbf{F}) \...
ConvexHull's user avatar
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Integral of differential $n$-form over half $n$-sphere

I'm struggling a lot with an exercise from my class. If $M=\{x \in \mathbb{R}^n: |x| = 1, x_n \ge 0\}$, then how can we calculate $\int_M \omega$, where $\omega = x_n \ dx_2\wedge...\wedge dx_n$? I ...
Diogo Santos's user avatar
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Equivalent divergence-free vector

I have a vector field F = $\frac{1}{(x^2+y^2+\frac{\beta^2}{4})^{\textrm{1/2}}}(-xi-yj -\frac{\beta}{2}k)$, where $\beta = \frac{4\pi}{\lambda}A\sin(\frac{2\pi}{\lambda}z)\left[A\cos(\frac{2\pi}{\...
MrSixStrings's user avatar
3 votes
2 answers
160 views

Problem with bounds on surface integral.

Let $$S=\{(x,y,z)\in\mathbb{R}^3:z=xy\},$$ and consider the $1$-form of $\mathbb{R}^3$ given by $$\omega=(y^3+xz)dx-x^2dz.$$ I'm trying to compute the integral $\int_C \omega,$ where $C=S\cap \{x^2+y^...
Guillermo García Sáez's user avatar
2 votes
2 answers
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Evaluate integral of $x^3dy \wedge dz+y^3dz \wedge dx+z^3 dx \wedge dy$ over the upper hemisphere.

Calculate the integral of the differential form $\omega$ over the half-sphere $S^{+}$, where: $\omega = x^3 dy \wedge dz + y^3 dz \wedge dx + z^3 dx \wedge dy $ $S^{+} = \Big\{ (x,y,z) \in \mathbb{R}^...
thefool's user avatar
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Integral of a 2-form on a smoothly parametrized surface

I am studying Stokes' Theorem and its applications, and dealing with the following two problems: Consider the surface $ S ={(x, y, z) \mid z=1-x^{2}-y^{2}>0}$. Find a smooth parametrisation $H$ of ...
autodidacti's user avatar
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Verifying the Stokes' Theorem

Given $$z=10-x^2-y^2,\quad1\leq z\leq9,$$ I have to verify the Stokes' Theorem with $$F(x,y,z)=(3z,4x,2y)$$ Since $\nabla\times F=(2,3,4)$ and as we can parametrize the surface as follows $$\sigma(r,\...
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Which boundary curves should I choose in Stokes' Theorem?

I have to verify Stokes' Theorem to the field $$F=(3z,4x,2y)$$ and the surface $$z=10-x^2-y^2,\quad 1\leq z\leq9$$ For $z=1$ and $z=9$ I get two circles. I'd like to know which one I should choose to ...
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If $\vec F=\frac{\vec r}{r^3}$ then $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {0} \to \mathbb{R}^3$ such that $\vec F = curl \,\vec G$.

If $\vec F=\frac{\vec r}{r^3}$ then show that $\textrm{div} \vec F = 0$ but $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {(0,0,0)} \to \mathbb{R}^3$ such that $\vec F = \textrm{curl} \,\vec G$. I can show ...
user1942348's user avatar
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Why is it that $\partial S^2=\emptyset$ [duplicate]

While doing some differential geometry I came across the following application of the Stokes Theorem. $$\int_{S^2}d(xdy)=\int_{\partial S^2}xdy$$ Now, the professor baldly affirms that $\partial S^2=\...
HornyPigeon54's user avatar
2 votes
1 answer
270 views

Stokes' theorem on a triangle

I've been given a question that I'm having trouble figuring out: Calculate \begin{equation} \oint_{T} xydx + yzdy + zxdz \end{equation} using Stokes' theorem, where $T$ is the triangle with vertices ...
WatT's user avatar
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2 answers
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Integration using Stokes' theorem

I have a problem that I can't seem to figure out. Given a surface $S$: \begin{cases} x^2+y^2 \leq 1 \newline z = y^2 \end{cases} Let $C$ be the edge of $S$. $C$ is oriented so that the projection of $...
WatT's user avatar
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1 answer
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How do we use the right hand rule for Stokes' theorem?

Let $C$ be the intersection curve between the plane $z = 10 - x - y$ and the cylinder $x^2+y^2 = 1$, oriented such that the projection of the curve onto the xy-plane is positively oriented. Determine ...
Need_MathHelp's user avatar
3 votes
1 answer
90 views

Stokes theorem applied to planes

Let $\textbf{F} = (y+z,-xz,y^2)$. Let $S$ be the surface above the $xy$ plane and bounded by $2x + z = 6$, $y = 2$, $y = 0$ and $x = 0$. Calculate $$\iint_S \text{curl } \textbf{F} \cdot d\textbf{S}$$ ...
James's user avatar
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1 vote
2 answers
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How to calculate the line integral when the vector field is conservative?

Calculate $$\int_{\gamma} x^3 dx + (x^3 + y^3)dy$$ where $\gamma$ is the line $y=2-x$ from $(0, 2)$ to $(2, 0)$. It was my understanding that if the vector field was conservative then the path doesn't ...
Need_MathHelp's user avatar
3 votes
1 answer
93 views

Calculate flow integral using Stokes's theorem

Calculate the flow integral $$\iint_{Y} \text{curl} (\vec{F}) · \hat{N} dS$$ where $Y$ is part of the sphere $x^2 + y^2 + (z − 2)^2 = 8$ that lies above the $xy$-plane and $\hat{N}$ is the outward ...
Need_MathHelp's user avatar
3 votes
0 answers
154 views

Is there a connection between shoelace formula and Stokes theorem?

The shoelace-formula is a method to calculate the area of a polygon. It is given as $$ A = 1/2 \sum_i{(x_i-x_{i+1})*(y_i+y_{i+1})} $$ for cyclical $i$. Expanding the product yields the terms $x_i y_i -...
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