# Stokes' Theorem ( Surface Integral )

• bfr
In summary, the problem involves using Stokes' theorem to find the value of a surface integral over a specific surface. The surface in question is part of a plane and the function used is f=(2z,-x,x). After some calculations, it is determined that the best approach is to use x = cos(theta), y = sin(theta), z = sin(theta)+1 and then apply Stokes' theorem. The resulting integral is -4π, giving the final answer.
bfr
[SOLVED] Stokes' Theorem ( Surface Integral )

## Homework Statement

Use stokes' theorem to find the value of the surface integral $$\int\int$$ (curl f) dot n) dS over the surface S:

Let S by the part of the plane z=y+1 above the disk x^2+y^2<=1, and let f=(2z,-x,x).

## Homework Equations

http://img187.imageshack.us/img187/291/1fdf437d8e18a23191b63dfnj8.png

## The Attempt at a Solution

So, S has a minimum at (0,-1,0) and a maximum at (0,1,2). This gives me a right triangle, with one of its legs from (0,-1,0) to (0,1,0), and the other from (0,1,0) to (0,1,2). Its hypotenuse is 2sqrt(2) units long, meaning the slanted elliptical surface has a long axis length of 2sqrt(2) and a short axis length of 2 (from (-1,0,1) to (1,0,1)).

Now, should I somehow try to calculate the circulation around this region using stokes's theorem (maybe have a z component of the path being some sort of trigonometric function?), or is there a better way to do this (by still using Stokes' theorem, though)?

EDIT: Nevermind. The best way is to just use x = cos(theta) , y = sin(theta), z = sin(theta)+1 after applying stoke's theorem.

Last edited by a moderator:
The integral then becomes\int_0^{2\pi} \nabla \times f \cdot n ds = \int_0^{2\pi}(-3cos(theta),3sin(theta),1) \cdot (cos(theta),sin(theta),cos(theta)) ds = \int_0^{2\pi} -2sin(2\theta) ds = -4\pi.

## What is Stokes' Theorem?

Stokes' Theorem is a mathematical theorem that relates the surface integral of a vector field over a surface to the line integral of the vector field around the boundary of the surface. It is named after Irish mathematician Sir George Stokes.

## What is the significance of Stokes' Theorem?

Stokes' Theorem is significant in many branches of mathematics, particularly in the study of vector calculus and differential geometry. It is used to evaluate complicated surface integrals by reducing them to simpler line integrals, making it a powerful tool in solving many physical and mathematical problems.

## What are the conditions for applying Stokes' Theorem?

Stokes' Theorem can only be applied to a closed surface, meaning that the surface is completely enclosed with no holes or gaps. Additionally, the surface must be smooth and continuously differentiable, and the vector field must be continuously differentiable on the surface.

## Can Stokes' Theorem be applied to any type of surface?

No, Stokes' Theorem can only be applied to surfaces that can be parameterized. This means that the surface can be described by a set of equations or functions that define the position of each point on the surface in terms of two parameters.

## What are some real-life applications of Stokes' Theorem?

Stokes' Theorem has numerous applications in physics and engineering, particularly in fluid dynamics and electromagnetism. It is used to calculate the flow of fluids through surfaces, as well as the circulation of electric and magnetic fields around closed surfaces. It is also used in the study of fluid flow in pipes and ducts, and in the analysis of ocean currents and wind patterns.

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