How Do Stokes' and Divergence Theorems Apply to a Cube's Surface Integral?

[jaejoon89]

Homework Statement

Given F = xyz i + (y^2 + 1) j + z^3 k
Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using
a) the divergence theorem
b) using Stokes' theorem

Homework Equations

Divergence theorem:
∫∫∫∇.FdV = ∫∫∇.ndS

Stokes theorem:
∫∫(∇xF).n dS = ∫F.dR

The Attempt at a Solution

The divergence theorem gives a dot product. Here we're asked for the cross product
∫∫(∇xF).n dS
but the divergence of the curl will be 0. The Stokes theorem applied here is nonzero. What's wrong?

 

[Billy Bob]

Stokes' Theorem also gives 0. There are six faces on the cube. Each face contributes one line integral around its boundary, or equivalently four line integrals across directed line segments. You can save yourself a lot of computation if you will draw a medium or large cube. Then start drawing little arrows to indicate each directed line segment edge, and continue drawing until you see what happens.