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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Flux of vector field. Stokes' Theorem

Let´s consider the vector field f $(x,y,z) = (y-z,z-x,x-y)$ and $ b > a >0 $ . How could we calculate the flux of f through the surface $S = \{x^2+y^2+z^2 = 2ax, x^2+y^2 \leq 2bx, z \geq 0\} $ ...
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Stokes theorem for integrating a scalar times normal over a surface area

I have the following formula in 3-Dimensions: $$ \int_{\partial \omega} f(x,y,z) \vec{dS} = \int_{\omega} \nabla f dV$$ I want to write the above in the language of forms and derive it through stokes. ...
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Calculating the flux of an open surface when the divergence is zero

When the divergence of a vector field thrugh an open surface is zero, does that mean that the flux through the surface is also zero? if not how do we then proceed with the stokes divergence theorem ...
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Should a compactly supported field have a Helmotz decomposition that is compactly supported?

Let $\bf F$ be a smooth vector field, which is null outside a finite compact domain $V$. By Helmoltz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} ...
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Transforming every volume integral into a surface integral

Helmotz decomposition theorem says, on one hand, that every vector field $F$ sufficiently smooth can be decomposed into the sum of a solenoidal field $\nabla\times \bf A$ and a gradient field $\nabla \...
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Verifying Stokes' Theorem with $\omega = x^2 \text{d}w-2yz \text{d}x$ as a one-form in $\mathbb{R}^4$ over a manifold.

I'm having trouble verifying Stokes' Theorem, and I'm just wondering if someone's able to find my mistake in the calculation. Let $\omega=x^2\text{d}w-2yz \text{d}x$ be a one-form on $\mathbb{R}^4$, ...
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Calculate line integral of plane intersecting with sphere

I saw a few posts on this sort of questions but I couldn't apply any idea on my particular question. Question : Calculate $\int_C (2z-x)dx+(2x-y)dy+(2y-z)dz$ when $C$ is the intersection of sphere $x^...
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Volume to surface integral of $R\times \nabla \times B$

I need to transform the following integral into a surface integral (if that's possible): $$\int\int\int_\Omega R\times (\nabla \times A) dv = \int\int_{\partial \Omega} ? . {\bf n} da, $$ where $R = (...
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Integrate 2-Form over the boundary of the standard cube

Find the integral of the form $dy∧dz+dz∧dx$ over the boundary of the standard cube in $R^3$. My thought: consider $\omega =y dz+ z dx$ so that $d\omega =dy∧dz+dz∧dx$ That is by Stokes, we have $$ \...
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How do I find the loop passing through certain points which maximizes a one form integral?

Suppose I have some points in a plane, and I want to integrate a form over a loop passing through all those points, how would I choose out of such loops the one which extremizes the integral? ...
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Stokes Question (non-conservative vector field but 0 flux as answer?)

I have a Stokes question. $C$ is a curve created by intersection of a cylinder $x^2+y^2=9$ and a plane $z=1+y-2x$. The curve is clockwise when viewed from the positive $z$-axis. The vector function ...
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Evaluate the surface integral over a cone region

Let $\omega = (xyz)dx +(xy^2)dy + (e^x)dz$ Evaluate $\int_S d\omega$ where $S$ is the cone with base $x^2+y^2 = 9$ in the xy plane and vertex at $(-2,3,1)$ oriented by the normal which points away ...
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Why is Green's theorem a special case of Stokes theorem?

I have already seen related questions and don't understand. Please help me. $\oint_C \mathbf{A} \cdot d\mathbf{r} = \iint_S (\nabla x \mathbf{A})\cdot \mathbf{n}$ dS Let $A \leq P,Q,0>$ Then $\...
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Evaluate the line integral over the curve of intersection [closed]

Evaluate $\int_c \frac{y^2}{2}dx + zdy + xdz$, where $c$ is the curve of intersection of the plane $x+z = 1$ and ellipsoid $x^2+2y^2 + z^2 = 1$. The keyword "intersection" guided me to set ...
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Verification of stokes' theorem on unit sphere cut by 2 horizontal planes

I have a region R = {$(x,y,z): x^2 + y^2 + z^2 =1, cosB \leq z \leq cos A$ }, where $ 0 \lt A \lt B \lt \pi.$ I am given a vector field in cylindrical polar coordiantes: $\vec F$ = f(z) $\vec e_{\...
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Physical Applications of Stokes' Theorem.

I'm currently helping an undergrad do readings on differntial geometry. The student is an engineering student, and I want to explore applications of Stokes' Theorem in physics/engineering. I don't ...
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Extension of flux to higher dimensions

I'm UG physics student. So sorry if this question is dumb. We are in phase space of $6N$ dimensions. Each point $ \mathbf r$ in this space has $6N$ coordinates. Pathria, Statistical mechanics writes ...
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Stoke's Theorem verification with $S$ as a Square.

The Vector field $F$ is given by $\mathbf{F}=\left\langle e^{y-z}, 0,0\right\rangle$. Consider the square $S$ with vertices $(9,0,4),(9,9,4),(0,9,4) \text {, and }(0,0,4) \text {. }$ I need to verify ...
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Stokes Theorem, integrating a "cut" region

I am having an issue with the two "Methods/Formulas" of Stoke's theorem. I am asked to evaluate a line integral. I have a curve C that is created by the intersection of the plane z=3-2x+y ...
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Wrong sign in the energy method on Minkowski space

Given any vector field $X$ in Minkowski space, we can use Stokes' theorem to derive $$ \int_{t_0}^{t_1}\text{div} X \text d V_g =\int_{\{t=t_0\}} g(X,\partial_t)-\int_{\{t=t_1\}} g(X,\partial_t), $$ ...
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Confusion in understanding Gauss Divergence theorem and Stokes theorem

Gauss divergence theorem says from https://en.wikipedia.org/wiki/Divergence_theorem that $$ \iiint_{V}(\nabla\cdot F)\,dV=\iint_{S}(F\cdot n)\,dS. $$ Now, let $G=\nabla\times F$ for a nice vector ...
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Green function Stokes equation

So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force, where $\textbf{P}=\textbf{P}(x,y,z)$,...
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Discontinuous vector field with curl 0

Let $S$ be a part of the paraboloid $z=1-x^2-y^2$ such that $z\geq 2|y|$. They ask to calculate $$ \int_C\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy+\frac{1+e^z}{1+z^2}dz $$ where the curve $C$ is ...
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Application of stokes theorem?

I was reviewing some class notes and I found the following problem: Let $$F(x,y,z)=\left(\frac{-yz}{x^2+y^2},\frac{xz}{x^2+y^2},e^z\right)$$ and let $\gamma:\left\{\begin{array}{rcl} z+1&=&x^2+...
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Stokes' theorem for a manifold without boundary

Stokes' theorem states that when $M$ is a compact oriented $m$-manifold with boundary, and $\omega$ is a $(m-1)$-form on $M$, we have $$\int_{\partial M}\omega = \int_{M} d\omega.$$ This is confusing ...
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Multivariable Calculus Exam Mistake?

This question was from an exam taken in January 2022 on a course on introductory multivariable calculus and was worded exactly as follows: "For a general surface $S$ bounded by a closed curve $C$ ...
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A Surface integral over the intersection of a plane and a sphere

I've been banging my head against this thing for the last 4 hours to no avail Evaluate the integral $ \iint_S curl \vec{F}\cdot d\vec{S}$ where $S$ is the portion of the surface of the sphere defined ...
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For Stoke’s theorem on Manifolds how is the total change over the boundary generalized

Stoke’s theorem is that the total change over the boundary is equal to the sum/integral of the derivatives over the region. Written out like this: $$ \int_D df = \int_{\partial D}f$$ If D is the ...
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Why is the exterior product an antiderivation intuitively?

I am attempting to motivate the form of the exterior derivative in the study of differential forms. In particular, I am trying to see why the following definition is compatible with Stokes' Theorem: ...
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Calculate the Solid Angle using Stokes' theorem

The solid angle for the surface S subtended at a point P is: $$ \Omega=\iint_{S} \frac{\hat{r} \cdot \hat{n}}{r^{2}} d S $$ where $\hat{r}$ and $\hat{n}$ are unit vectors and $r =|\vec {r}|$ is the ...
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Can someone help me in understanding how to apply stokes theorem here?

Quick remark, the question couldn't be answered at this point so I'm happy go read more answers I have some problems in understanding how to apply stokes theorem. So we had the following version: Let ...
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Calculate Integral of intersection between two surfaces

Calculate $$\int_{\partial F} 2xy\, dx + x^2 \, dy + (1+x-z) \, dz$$ for the intersection of $z=x^2+y^2$ and $2x+2y+z=7$. Go clockwise with respect to the origin. My attempt: I first calculate the ...
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Stokes' theorem on a paraboloid

I am currently undertaking a class on Classical Electrodynamics, and I am realising my multivariable calculus skills are slightly rusty, as it has been a few years. Anyhow, I am given a surface $$S\{(...
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Orientation of line - Stoke's Theorem

The question states: Surface $S$ is the part of the cone $y = \sqrt{x^2+y^2}$ with $0 \leq y \leq 3$ oriented in the direction of the positive y-axis. Consider the vector field $F(x,y,z,) = <5yz,...
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Need help computing flux integral of vector field over an unclosed shape

I need help computing this integral: $$ \iint_{S} \frac{x \hat{\imath} + y \hat{\jmath} + z \hat{k} }{ ( x^2 + y^2 + z^2 )^{3/2} } \cdot \hat{n} \, dS$$ over S that's defined as: $$S = \...
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4 votes
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Stokes' theorem in higher dimensions

I'm trying to follow along with Stokes's theorem on page 46 and I'm not quite understanding where this comes from: $$ V=\int_{B} dx \wedge dy \wedge dz \wedge dt = - \int_{S} t dx \wedge dy \wedge dz $...
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Stokes Theorem without compact support

I am looking for a version of Stokes Theorem on manifolds that does not assume compact support of the differential form $\omega$. In fact I have found the following statement in the book Riemannian ...
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Evaluating line integral using The Stokes' Theorem

I want to evaluate $$\oint_C (x-z) dx + (x + y) dy + (y+z) dz$$ where $C$ is the ellipse, in which the plane $z=y$ intersects the cylinder $x^2 + y^2 = 1$, oriented counterclockwise as viewed from ...
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A problem about the area element in the Stokes' Theorem

Given a vector field $F(x,y,z) = x^2 \hat i + 2x \hat{j} + z^2 \hat{k} $ and a curve $C: \text{the ellipse } 4x^2 + y^2 = 4 \text{ in the } xy- \text{plane}$, I want to find $$\oint_{C} \vec F \cdot ...
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Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$

Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$ oriented counter clockwise from xy plane First, I made the following parametricisation: ...
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Calculate the circulation of the vector field alone a parameterized circle (Stoke's Theorem...?)

Find the circulation of the following vector field $\vec{F}(x, y, z) = \langle \sin(x^2+z)-2yz, 2xz + \sin(y^2+z), \sin(x^2+y^2)\rangle$ along the circle $\vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle$ ...
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Interval of stokes theorem

I have the vectorfield $F=(y e^x,x^2+e^x,z^2)$ and the curve $r(t)=(1+cos(t),1+sin(t),1-cos(t)-sin(t))$ I have to find the line integral of this using stokes theorem, however in this case I'm only ...
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Deriving boundary conditions for Faraday's law and Ampére's law by letting the width $\delta$ approach zero

I am currently studying the textbook Physics of Photonic Devices, second edition, by Shun Lien Chuang. Section 2.1.1 Maxwell's Equations in MKS Units says the following: The well-known Maxwell's ...
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How would you discover Stokes's theorem?

Let $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$. Stokes's theorem ...
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Divergence of curl is zero (coordinate free approach)

I'm TAing a vector calculus course and the professor has asked the students to prove that $\nabla \cdot (\nabla \times \vec{F}) = 0$. I'm meant to teach this problem in recitation tomorrow and I think ...
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Proof of the fact that one-form $\omega$ is exact $\Rightarrow$ $\int_{S^1} \omega =0.$

I saw the proof of $$\mathrm{one-form} \ \omega \ \mathrm{is \ exact} \Rightarrow \int_{S^1} \omega =0$$ where $S^1$ is an unit circle. The proof is here. Since $\omega$ is exact, there exists $f\...
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Help with visualizing the surface to be able to calculate the line integral

We are given the vector field $F(x,y,z)=[2z+y, 2x+z, 2ycos(z)]$ A surface S is composed of two parts. One part $S_1$ given by $z=x^2+y^2$, for $0\leq z\leq4$ and $0 \leq y$. The second part $S_2$ is ...
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5 votes
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"Improper" Stokes theorem

In this question I will use Stokes' theorem with manifolds (well, domains in manifolds), rather than with chains. Let $M$ be a smooth manifold of dimension $m$. A subset $D\subseteq M$ is called a ...
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Metric independent definition of the derivative

In the wikipedia article on the exterior derivative it says: "[The exterior derivative] allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's ...
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2 answers
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$\int_c 3y dx +5x dy + \frac{2x+3}{z^2} dz$ for the intersection between two surfaces

For the closed curve, $c = \{(x,y,z) \vert x^2 + y^2 - z^2 =0\} \cap \{(x,y,z) \vert (x-1)^2 + y^2 =4\}$ Find the $\int_c 3y dx +5x dy + \frac{2x+3}{x^2 + y^2} dz$ First I focused the surface $\{(x,y,...
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