# Stokes Theorem for Surface S: Parametrization, Flux and Integral

• JaysFan31
In summary, the conversation discusses the parametrization of a surface S, which is swept out by a line segment joining two points and is bounded by a curve A. The vector field F and its flux through the surface S are also mentioned, along with the computation of the integral for the force field F and the boundary curve A. The conversation highlights the use of Stokes Theorem and the difficulty in setting up the integrals due to the lack of an equation for the surface S. The parameter r is used to sweep out the line segment and the line segment itself plays a role in determining the bounds of integration. The conversation also mentions the need to find the unit normal n, which is dependent on the line segment joining the two points.
JaysFan31
For the surface S (helicoid or spiral ramp) swept out by the line segment joining the point (2t, cost, sint) to (2t,0,0) where 0 is less than or equal to t less than or equal to pi.

(a) Find a parametrisation for this surface S and of the boundary A of this surface.

I can only guess that
x=rcost
y=rsint
z=t
0 less than or equal to t less than or equal to pi
0 less than or equal to r less than or equal to 1
Is this right or totally bogus?

(b) For the vector field F=(x,y,z) compute the flux of F through the surface S. Assume the normal to the surface has a non-negative k component at t=0.
No idea because what is the surface S? It has no equation.

(c) Compute integral (F*dr) where A is the boundary curve of the surface S and F is the force field (x,y,z).
I think I just use Stokes Theorem for (c), but I'm having trouble setting it up since again I have no equation for the surface S. I also don't know how the whole line segment joining works into it.

I can evaluate the integrals, I just have trouble setting them up. If anyone could help me with these three I would appreciate it. Just explain what's going on.

You have just given a parametric form for the surface, x(r,t)=r*cos(t), y(r,t)=r*sin(t),
z(r,t)=t. How can you say S has no equation?

Well is that right for a helicoid?
And how does the line segment joining those two points play into the helicoid?

The parameter r sweeps out the line segment. r=0 and you're on the z-axis, r=1 and you are on the helix. r=in between and you are in between.

So does it factor into the bounds of integration?

I guess I don't see what the 2t means for the x part also?

It just seems when I do the integrations out, it's going to be simply
t between 0 and pi
r between 0 and 1
for both parts (b) and (c) if I use parametrisation.

Does the line segment joining the point (2t, cost, sint) and (2t,0,0) somehow give me the unit normal n which I have to find?

## 1. What is Stokes Theorem for Surface S?

Stokes Theorem for Surface S is a mathematical theorem that relates the surface integral of a vector field over a surface S to the line integral of the same vector field over the boundary curve of S. It is a fundamental result in vector calculus and is used to solve problems related to fluid flow, electromagnetism, and other physical phenomena.

## 2. How is Surface S parametrized?

Surface S is typically parametrized using two parameters, u and v, which define the coordinates on the surface. These parameters are used to generate a set of points on the surface, which can then be used to calculate the surface integral in Stokes Theorem. The specific parametrization used will depend on the shape and orientation of the surface S.

## 3. What is the flux in Stokes Theorem for Surface S?

The flux in Stokes Theorem for Surface S represents the amount of flow of a vector field through the surface S. It is calculated by taking the dot product of the vector field and the unit normal vector to the surface at each point, and then integrating over the entire surface. The flux is a measure of the strength and direction of the vector field passing through the surface.

## 4. How is the integral calculated in Stokes Theorem for Surface S?

The integral in Stokes Theorem for Surface S is calculated by first parametrizing the surface S and then evaluating the surface integral using the parametrization. This involves taking the dot product of the vector field and the surface normal, and then integrating over the surface with respect to the two parameters, u and v. The resulting integral will depend on the specific parametrization and bounds chosen for the parameters.

## 5. What are some real-world applications of Stokes Theorem for Surface S?

Stokes Theorem for Surface S has many real-world applications in fields such as fluid dynamics, electromagnetism, and engineering. For example, it can be used to calculate the flow of a fluid through a curved pipe, the magnetic field around a wire, or the stress and strain on a curved surface. It is an essential tool for solving problems involving vector fields and surfaces in various scientific and engineering disciplines.

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