I was tasked with calculating $$\oint_{L}F\mathrm{d}r$$ for when $F=xz\mathbf{i}-\mathbf{j}+y\mathbf{k}$ (vector form) and $$L = \begin{cases}z=5(x^2+y^2)-1 & \mbox{ } \mbox{} \\z=4 & \mbox{} \mbox{} \end{cases}$$
Using:
- The Stokes Theorem
- Directly
At 1) I did manage to calculate the curl which is $\langle 1,x,0\rangle $ but I cannot go on with the calculation as I do not know what $\mathrm{d}S$ is.
As for 2) I have no idea.
Any help?
$\mathbf{dS}$ is used for the surface integral. The link is dense; the important part are the equations $$ \vec n = \frac{\vec r_u\times\vec r_v}{\|\vec r_u\times\vec r_v\|}\quad\text{and}\quad\iint_S \vec F\cdot d\vec S= \iint_S\vec F\cdot \vec n\,dS $$
Basically, you need to parameterize the surface and then do a double integral over it.
The story is the same for the second: parameterize the circle where the paraboloid intersects the plane $z=4$. I'd look at example 1 here. You can refer to the way they parameterize $r$, since the shape is very similar to the one you're working with.
There are a lot of basic concepts that you need to be familiar with when working with Stokes', Green's, and the divergence theorem. Tutorial.math.lamar is a good place to read up on these (the links I keep referencing).