Mathematics 241 -- Multivariable Calculus
Review Sheet for Exam 2
October 19, 2007
General Information
The second exam for the course will be given Friday, October 26,
as announced in the course syllabus. It will cover the material
we have discussed in class (including the labs)
through and including the material on the del operator and divergence
and curl of vector fields from
from class on Friday, October 19.
The topics are covered in sections 2.3, 2.4, 2.5, 2.6, 3.1, 3.2, 3.3, 3.4
of the text. (But note that there is quite a bit more in the text than we
have discussed in class. You are only responsible for things we have done
in class.) Here's a breakdown by topic:
- The tangent plane to a graph z = f(x,y) at
a point (x,y,z) = (a,b,f(a,b)) and how that relates to the concept of
differentiability of f(x,y). Also, know the theorem
that if f has continuous partial derivatives on some region, then
f is differentiable on that region.
- The derivative matrix for F : Rn -> Rm.
- Higher order partial derivatives.
- The multivariable chain rule (both the matrix form and the
component-by-component forms).
- Directional derivatives and the gradient.
- Tangent vectors and lines, arclength, the T,N,B vectors, curvature, and torsion for parametric curves in R3.
- Vector fields and flow lines. The gradient as a vector field.
- The del operator, divergence and curl for vector fields.
There will be 4 or 5 problems, each with several parts. Some may
ask for a graph or the result of a calculation; others may ask for
a description or explanation of some phenomenon (similar to some
questions from labs).
Important Note
Scheduling a review session before the exam this time is going
to be tricky: I have off-campus commitments Tuesday and Thursday
evenings, and an exam for another class starting at 6:00pm on
Wednesday. Earlier in the afternoon on Wednesday would be a possibility.
Suggested Review Problems
For practice, it would be a very good idea to
try a few of the odd-numbered problems from all the sections
of the text that we have covered. Here are some good ones:
Section 2.3/9, 11, 15, 21, 23, 29.
Section 2.4/11, 13, 17.
Section 2.5/7, 11, 13.
Section 2.6/5, 17, 21.
Section 3.1/9, 15, 19, 27.
Section 3.2/5, 15, 17 (but in part a, use the parametrization
α(t) = (t, f(t),0) and our usual method to find the curvature)
Section 3.3/3, 5, 19, 21, 23.
Section 3.4/1, 3, 7, 9, 14, 15.
Sample Exam Questions
Note: the actual exam will not be this long. Also, the actual
exam questions may ask things in different ways and may involve
examples with different properties.
I.
- A) Let w = sin(st) cos(st). Show
that s (∂ w/∂ s) = t (∂ w/∂ t).
- B) More generally, if w = f(x)
is any function and x = st, then show using the Chain
Rule that s (∂ w/∂ s) = t (∂ w/∂ t).
II. Let f(x,y) = ex - y and g(s,t) = (s2 - t2,
s t + 2t)
- A) True or false, and give a reason: f
is differentiable at all (x,y) in R2.
- B) Compute the derivative matrices for f and g.
- C) Where does the tangent plane to z = ex - y at (1,1,1)
meet the z-axis?
- D) Find the directional derivative of f(x,y) at a = (2,1)
in the direction of the vector v = (1,3) (Note: (1,3) is not a
unit vector!)
- E) Use the Chain Rule to find the derivative of the composition (f o g)
at (s,t) = (3,0).
- F) Check your answer in part E by computing the composition and
differentiating directly.
III. A mountain has the shape of the graph z = c - a x2 - b y2,
where a,b,c are positive constants. Here x,y are the east-west
and north-south map directions respectively, and z is the altitude
above sea level.
- A) At (x,y) = (1,1), in what direction is the altitude increasing
the most rapidly?
- B) If a marble was released from (1, 1, c - a - b) on the hill,
in which direction would it begin to roll?
- C) If you were hiking along the hill starting from (1, 1, c - a - b),
in which direction should you move at first to remain at a constant altitude?
IV. Let z = f(x,y), where f has continuous partial derivatives.
If we change coordinates to polar coordinates x = r cos(θ), y = r sin(θ),
show that the partial derivatives of z satisfy:
(∂ z/∂ x)2 + (∂ z/∂ y)2 =
(∂ z/∂ r)2 + (1/r2) (∂ z/∂ θ)2.
V. Let x(t) = (sin(3t), cos(3t), 2t3/2).
- A) Compute the arclength of the segment of this curve from t = 0 to
t = 1.
- B) Find the unit tangent vector T at a general point on the curve.
- C) An alternative formula for the curvature is
κ(t) = ||x'(t) x x''(t)||/||x'(t)||3 (cross product
of velocity and acceleration in the numerator). Use this
to determine the value of the curvature at t = π
for this curve.
VI. Let F(x,y,z) = (yz, xz, xy) be a vector field on R3.
- A) Show that F(x,y,z) = (0,0,0) at any point on one of the
three coordinate axes in R3.
- B) Compute the divergence of F.
- C) Compute the curl of F.
- D) Find a scalar function f such that F = ∇ f.
- E) Show that α(t) = (1/(1-t), 1/(1-t), 1/(1-t)) is a flow
line of F.