Math 320 Sixth Test ______________________
Calculus III Name
This 50 minute test covers sections 16.4-9 of "Calculus" by Stewart. Each part of each problem
on this unexpectedly easy first page is worth two points. All other problems are ten points each
(unless otherwise indicated).
1. Fill in the blank with the letter of the answer which most closely matches.
∇f a. indicates the "flow out of " points
∇ . F b. indicates the "circulation around" points
∇xF c. indicates the rate and direction increase at points
2. True or False? Place T for true or F for false in the blanks below.
∇ = ∂
∂x i + ∂
∂y j + ∂
∂z k is the "del" operator.
To apply Stokes theorem and the divergence theorem the surface must be
oreintable.
The Möbius strip is an oreintable surface.
∇ x ∇f = 0 (when f has continuous second order partials).
T ∇ ∇xF = 0 (when F has continuous second order partials).
3. Place a T in the blanks below if the property is equivalent to the field F being conservative
on an open simply-connected region D in space. Otherwise place an F.
⌡
⌠
O
C F.dr = 0 for every piecewise smooth closed path C in D.
F.dr is an exact differential form.
⌡
⌠
b
aFdr is independent of path.
F = ∇f for some scalar function f.
There exists a potential function f on D for which ⌡
⌠
b
aFdr = f(b) − f(a).
∇ x F = 0 on the region D.
