Test 2 Review Sheet—Math 222
Prof. Frank, Spring 2005
This exam will cover sections 4.1-4.4, 5.1-5.5, 6.1, 6.2, 7.1, 7.2 (13 total).
General principles to guide your study:
(1) Review your notes for each section of the book, and the comments in the study guide for
that section. Make a note of every important fact, definition, and theorem from that section
that you feel you should memorize.
(2) Go over your HW assignments and make sure that you understand all of the problems, as
well as the related problems in the text. Pay close attention to the ones you missed the
first time around.
(3) Think about the questions listed below after you have completed your review. Try and
figure them out without referring to your notes.
(Warning: some of the following questions are much more vague and open-ended than what will
be on your exam. Consider them food for thought.)
(1) What is the difference between real-valued functions and vector-valued functions? What
are the two main types of vector-valued functions we’ve studied, and how can you graph
them?
(2) Suppose that cis a path for which c00 = 0. Is there a force acting on c?
(3) Suppose cis a unit speed path. What does this mean? Define the vectors in the Frenet
Frame of c.
(4) Suppose cis a unit speed path. Is it possible for B(t)6=0but N(t) = 0? Explain twice:
once with mathematical expressions and again on an intuitive level.
(5) Prove that the vectors in the Frenet Frame must be othogonal to their derivatives.
(6) Define the curvature function k(t) and the torsion function τ(t). Explain what they tell
you about your unit speed curve.
(7) If c(t) is a unit speed path with zero curvature, what does this imply about c? If instead
it has zero torsion, what does that imply about c?
(8) What does it mean to say that Fis a gradient vector field? Is every vector field a gradient
vector field? Either prove or find an example to the contrary.
(9) Suppose F:R3→R3is a vector field and that cis a flow line for F. Give the domain and
range of c, and the precise equation connecting it to F. Explain the equation in a sentence.
(10) True or false? Give a proof if true and a counterexample if false.
(a) If the divergence of a vector field is zero, then the curl must be nonzero.
(b) If the divergence of a vector field is zero, then the curl must be zero also.
(c) If Fis a C1gradient vector field, its divergence is zero.
(d) If Fis a C1gradient vector field, its curl is zero.
(11) Compare and contrast the methods of obtaining volumes via Riemann sums and Cavalieri’s
principle. Draw pictures to illustrate your discussion.
(12) Explain the two things in Fubini’s theorem that make computing double and triple integrals
easier.
(13) Let D= [a, b]×[c, d]. Show that ZZD
∂2f
∂x∂ y dA =f(a, c)−f(b, c) + f(b, d)−f(a, d).
(14) Let Dbe the region given by x2+y2<4. Without doing any calculations, explain why
ZZD
xdA = 0.
