Calculating Curve Integrals with the Del Operator: A Pain in the Brain?

[Karl Karlsson]

Homework Statement

Consider the vector field
$\vec v = exp(\frac {xy} {r_0^2}) [\frac {z} {r_0^2} (x\vec e_1 + y\vec e_2) + \vec e_3] + \frac {1} {r_0} (x\vec e_2 - y\vec e_1) + cos(\frac {z} {r_0})\vec e_3$
where $r_0$ is a constant with dimension length.

a) Calculate $\nabla\cdot\vec v$ and $\nabla\times\vec v$.
b)Calculate the circulation integral $$\oint_Γ \vec v \,d\vec S$$ where Γ is the curve that is parameterized by $x = r_0cos(t), y = r_0sin(t), z = r_0cos^2(2t), (1 < t < 2\pi)$

Relevant Equations

$\vec v = exp(\frac {xy} {r_0^2}) [\frac {z} {r_0^2} (x\vec e_1 + y\vec e_2) + \vec e_3] + \frac {1} {r_0} (x\vec e_2 - y\vec e_1) + cos(\frac {z} {r_0})\vec e_3$

My attempt is below. Could somebody please check if everything is correct?

Thanks in advance!

 

[Delta2]

This problem looks like it is setup to cause calculation pain in the brain .

Using wolfram I checked your answers for the divergence and the curl and I found them to be correct.

Cant find an easy way to check the curve integral. That calculation is really a pain in the brain...