Using stoke's theorem to calculate circulation

[charmmy]

Homework Statement

Use the surface integral in stoke's theorem to calculate the circulation of the filed F around the curve C in the indicated direction:

F= x²y³i + j+ zk
C; the intersection of the cylinder x²+y²=4 and the hemisphere
x²+y²+z²=16, z>=0, counterclockwise when viewed from above

Homework Equations

The Attempt at a Solution

Using stoke's theorem.
∫ ∫ curl F ⋅ n dS

I know how to compute the curl of F, but in this case how do I get the n? since it is an intersection of both surfaces? Do I need to equate them to each other and because x²+y²=4 ; correspondingly
x²+y²+z²=16 ==> becomes z^2=12?

 

[LCKurtz]

You have a choice of what surface to use which has the intersection curve as its boundary.

z² = 12 tells you that the intersection curve lies on the plane z = 2sqrt(3), so the intersection curve is just x²+y² = 4 on that plane. Now, the top portion of the sphere above the cylinder has that curve as its boundary, but so does the circle in the plane z = 2sqrt(3). The flux through either of those surfaces S satisfies

$$\int\int_S \nabla \times \vec F\cdot \hat n d S =\oint_C \vec F\cdot d\vec R$$

So use the flat planar disk with its normal direction given by the right hand rule (up in this case).

 

[charmmy]

So, essentially, the boundaries of the radius is determined by whichever surface that has the smaller diameter? In this case, we use 0<r<2 (of the cylinder) instead of 0< r < 4 (of the hemisphere which has a bigger radius)?

 

[LCKurtz]

So, essentially, the boundaries of the radius is determined by whichever surface that has the smaller diameter? In this case, we use 0<r<2 (of the cylinder) instead of 0< r < 4 (of the hemisphere which has a bigger radius)?

I wouldn't put it that way. Two surfaces will generally intersect in a curve. How you get the equation of the curve varies with the problem. Sometimes, as in this problem, you can get the equations by substituting values from one equation into the other. Other times you can find a parametric representation of the intersection curve. In this case the intersection curve is a circle of radius 2 in the plane z = 2sqrt(3). You can use any surface bounded by that circle to calculate the flux integral. The easiest choice is the circular disk in that plane.

 

[charmmy]

Thanks.. But to make it a bit more easy to understand, do you by any chance have any example where we have to find a parametric representation of the intersection curve, instead of being able to compute it directly by substitutions of equations? That would be of a great help!

 

[LCKurtz]

Thanks.. But to make it a bit more easy to understand, do you by any chance have any example where we have to find a parametric representation of the intersection curve, instead of being able to compute it directly by substitutions of equations? That would be of a great help!

OK, but you do realize in your current problem you don't want to do the circuit integral in the first place, right? You want to do the flux integral over the disc.

To answer your question, suppose you had a problem to integrate around the curve given by the intersection of the cylinder x²+y² = 9 and the slanted plane z-y=4. Since the cylinder is circular, the slanted plane will intersect it in a slanted ellipse. In this kind of problem it might be well to use the cylindrical (polar) angle θ and represent the curve parametrically in terms of it:

$$\vec R(\theta) = \langle 3\cos\theta, 3\sin\theta,4 +3\sin\theta \rangle$$