Test 2 Review Sheet Hints—Math 222
Prof. Frank, Spring 2005
(1) Real-valued functions look like f:Rn→R, while vector-valued functions look like f:
Rn→Rm, where m > 1. The two main types are paths (c:R→Rm) and vector fields
(F:Rm→Rm). Look in your notes for how to graph them.
(2) By Newton’s second law there is no force acting on it.
(3) It is a unit speed path if c0(t) = 1 for all t. In this case we have the tangent vector
T(t) = c0(t), the principal normal vector N(t) = T0(t)
||T0(t)||, and the binormal vector B(t) =
T(t)×N(t).
(4) It is not possible to have a nonzero binormal vector if the principal normal vector is 0.
Mathematically, this is by definition of the binormal as T×N: if N=0then B=0as
well. Intuitively this is because N(t) tells you how the direction of the tangent vector is
changing and B(t) tells you how the path is bending out of the plane of Tand N. If N=0,
then the direction is not changing and so obviously the path isn’t bending at all, especially
not out of the plane of Tand N.
(5) This is the trick that starts with the fact that ||T(t)|| = 1 implies ||T(t)||2= 1.
(6) The curvature function is given by k(t) = ||T0(t)|| =||c00(t)||. It is a measure of how much
the curve is bending in the plane given by Tand N(the osculating plane). The torsion
function is given by the equation B0(t) = τ(t)N(t), and is a measure of how much the curve
is bending out of the osculating plane.
(7) The curvature being zero means that the path is a line. The torsion being zero means that
the curve lies in a plane.
(8) It means there is an f:R3→Rsuch that F=∂f
∂x ,∂f
∂y ,∂f
∂z . It is an interesting point
that not every vector field is a gradient vector field. Find the example in your notes or just
make one up.
(9) We must have c: [a,b]⊆R→R3, and then we have that c0(t) = F(c(t)). The tangent to
the path always belongs to the vector field.
(10) (a) False. The zero vector field, F=0, is a trivial example.
(b) False. Any vector field of the form F(x, y)=(F1(y), F2(x)) has gradient 0 but will
usually have nonzero curl.
(c) False. Consider ∇fwhere f(x, y, z) = x2+y2+z2.
(d) True. If F=∇ffor some f:R3→R, then F=∂f
∂x ,∂f
∂y ,∂f
∂z . Set up the matrix to
compute the curl and then use that the mixed partials must be equal.
(11) In both methods, you are slicing the region into small pieces that estimate the volume, then
refining the slices to get better and better approximations, which in the limit result in the
exact volume. But the pieces are very different in the two cases. In Cavalieri’s principle
one slices the solid as if it were a loaf of bread, while Riemann sums slice the solid as if it
were french fries.
(12) Double and triple integrals are defined in terms of Riemann sums, but Fubini’s theorem tells
us that they can be computed as iterated integrals, which means we can use one-variable
integration techniques to compute them. Moreover, Fubini’s theorem says that we can
switch the order of integration if we want to.
(13) You can write Zb
aZd
c
∂
∂y ∂f
∂x dydx =Zb
a"∂f
∂x (x, y)
d
c#dx =Zb
a
∂f
∂x (x, d)−∂f
∂x (x, c)dx
=f(x, d)−f(x, c)
b
a=f(b, d)−f(a, d)−f(b, c) + f(a, c).
