Fall 2010 UBC

Important Review Problem

A very good review question for the final exam:

If $S_1$ is the upper hemisphere ($z \ge 0$) of a unit sphere oriented upward, and $S_2$ is the unit disk in the $xy$-plane oriented upward, is $\iint_{S_1} \mathbf{F} \cdot d\mathbf{S} = \iint_{S_2} \mathbf{F} \cdot d\mathbf{S}$?

What if I claimed that $\iint_{S_1} \mathbf{F} \cdot d\mathbf{S} > \iint_{S_2} \mathbf{F} \cdot d\mathbf{S}$.  Would you think I was lying?  If not, what could you conclude about the divergence of $\mathbf{F}$?

Answer in the comments if you like; I’ll post a further discussion tomorrow.

[EDIT:  Haven’t had any comments or questions about this one.  It is designed to remind you that you can’t always change one surface into another with the same boundary, but it requires that divergence is zero.]

Comments on: "Important Review Problem" (2)

1. Kyla Burrill said:

The original statement is only true if the vector field F is the curl of another vector field.

If F is the curl of another vector field, and C is the same boundary for S1 and S2 then according to Stokes’ theorem the original statement is true.

….in the right ball park?

2. That’s right; if F is a curl field, then the statement of equality is true. This uses Stokes’ Theorem.

Alternatively, you can also say that if F is incompressible (has divergence zero), and it is defined in the region between the two surfaces, then the statement of equality is true. This uses the Divergence Theorem.