How do you use Stokes' Theorem to calculate the surface integral over a cylinder of $\nabla \times F$? Do you have to calculate the line integrals along the top and the bottom? If so, is this example done incorrectly? Should the top line integral also be calculated? I don't understand why they only calculate the line integral in the $xy$ plane.
Also, if I do need to calculate both integrals, do I need to orient them differently?
If you want the actual problem, it is:
Use Stokes' Theorem to evaluate
$\int\int_T curl(xz \vec{j})d\vec{S}$,
where T is the cylinder
$x^2 + y^2 = 9$ with $0\leq z \leq 2$, orientated with an outward pointing normal.
But don't worry too much about the computation, I'm struggling more with the concept. I'm also pretty sure I could just do the integral without Stokes', but it's in the section on Stokes' theorem, so I should probably learn how to do it that way.
Best Answer
Yes, you do need to calculate both integrals and add them to get the answer.
However, you must be careful with the orientation of the curves.
Since the normal vector points outward($ \hat n = cos \phi \space \hat x \space + sin \phi \space \hat y$), the 2 curves $c_{top}$ and $c_{bot}$ that together make up the boundary of the cylinder, should inherit their orientation from the cylinder by applying the rule.
Rule:
Work for each curve seperately. Imagine yourself walking along the curve with your foot stepping on the curve and your head pointing as the normal vector indicates(i.e. radially outward). The orientation of the curve is chosen so that ,in your walk, you always have the surface to your left.
This image may help with the visualisation
Image: example for outward oriented open cylinder
This pdf helped me understand how to properly orient the curves of the boundary of an open cylinder, hope it helps someone else as well.
@page 7
http://www.owlnet.rice.edu/~fjones/chap13.pdf
I also found this relative topic.
Stokes' Theorem which boundaries to integrate