Integrating a complicated function using Stokes' Theorem

Question

$$\vec{\phi}(x,y,z)=\biggl(\frac{\ln(x^5y^3z^2+1)}{z^7+y^4+5}-yx^2e^{2z}, \ \ \cos^2(\pi z)xy^2, \ \ e^{x^2y^2}+\cos z\sin z^3 \biggr)$$

Then let $S$ be the upper-hemisphere $x^2+y^2+z^2=4$ with outward $\vec{n}$. Using Stokes' Theorem, find $$\iint_S(\nabla \ \times \ \vec{\phi})\cdot d\vec{S}$$

What do I do to make this easier to integrate?

Answer 1

Stokes' Theorem allows us to write surface integrals of the curl of a vector field $\nabla \times \vec{F}$ into line integrals of the vector field around the boundary, i.e., $\iint_S(\nabla\times\vec F)\cdot d\vec S = \oint_{\partial S} \vec F \cdot d \vec s$. Here the boundary of the hemisphere is the circle $C$ of radius $2$ in the $xy$- plane: $\{(x,y,0) \mid x^2 + y^2 =4 \}$. So we need to set up the line integral $\oint_C\vec \phi \cdot d \vec s$.

We can parameterize $C$ by $r(t) = (2\cos t , 2 \sin t, 0)$ for $t\in [0,2\pi]$. I'll show how to set up the integral, but I'll put a spoiler tag on it in case you want to try setting it up from here.

Setting up the integral:

We have that $r'(t) = (-2\sin t, 2\cos t, 0)$. So \begin{aligned}\oint_C \vec \phi \cdot d \vec s &= \int_0^{2\pi}\vec\phi(r(t)) \cdot r'(t)\, d t \\ &= \int_0^{2pi}(-(2\sin t) (\cos t)^2, (2\sin t)^2 (2\cos t), e^{16\cos^2 t \sin^2 t}) \cdot (-2\sin t, 2 \cos t, 0)\, dt \\ &= 2 \int_0^{2\pi}(4\sin t\cos t)^2 dt.\end{aligned} From here, we can use standard integration techniques and trig identities to evaluate the integral.