# 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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### 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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### 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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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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### 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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### "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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### $\int_c 3y dx +5x dy + \frac{2x+3}{z^2} dz$ 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,...