Name
Math 311-502
Final Examination
Instructions: Show all work in your bluebook.
R1
1. (25 pts.) Consider the real inner product defined on P2 by hp, qi ≡ 0 p(x)q(x)x2 dx.
Use the Gram-Schmidt process and this inner product to turn E = {1 + x, 1 − x, x2 } into
an orthogonal basis for P2 .
2. Let f (x, y) = (xy + y 2 , x − 2y, x2 − y 2 + 1) = (u, v, w) and also let g(u, v, w) =
(uvw, u2 − v 2 − w2 ) = (s, t).
(a) (10 pts.) Use the chain rule to find the affine approximation to g◦f (x, y) at x = y = 1.
(b) (5 pts.) Does a function inverse to g ◦ f (x, y) exist near x = 1, y = 1, s = −2, t = 2?
If so, what is its affine approximation?
3. (20 pts.) Let (x, y) = T (u, v) = (u, veu ), and let R be the u-v rectangle 0 ≤ u ≤ 1,
0 ≤ v ≤ 1. Sketch the image region T (R) and compute the double integral
Z
ye−2x dxdy .
T (R)
4. (20 pts.) Let F(x, y, z) = (2xyz + 1)i + (x2 z + z 2 )j + (x2 y + 2yz)k. Determine whether
or not F is conservative, i.e., whether F = ∇f for some scalar-valued function f . If it is
conservative, find f .
5. Consider the spiral curve γ parametrized by r(t) = a cos ti + a sin tj + btk.
(a) (10 pts.) Find the arclength parametrization of γ; that is, find r(s).
(b) (10 pts.) Find the tangent vector T(s) and the principal normal vector, N(s).
6. (20 pts.) Let F(x, y) = xy 2 i + 3x2 yj and let C be the curve gotten by traversing in
the counterclockwise direction
the the triangle with vertices (0, 0), (1, 1), and (0, 1). Use
H
Green’s Theorem to find C F · dx.
7. Let S be the surface of the sphere with center (0, 0, 0) and radius 2, and let n be the
outward drawn normal to S.
(a) (15 pts.) Let F = xy 2 i + yz 2 j + zx2 k, and let n be the unit normal to S. Set up,
but do not evaluate, an iterated double integral that gives the surface integral
ZZ
F · n dS .
S
(b) (15 pts.) Use the Divergence Theorem to convert the surface integral in part (a)
into a volume integral, and then evaluate the volume integral.
8. Consider the surface Σ (see Fig. 1) represented parameterically by
x2 + y 2
k,
r(x, y) = xi + yj +
2
where x2 + y 2 ≤ 4.
(a) (15 pts.) For the parametrization given, find both the standard normanl N and the
element of surface area dS. Also, find n, the unit normal to Σ shown in Fig. 1. Set
up, but do not evaluate, an iterated double integral that gives the area of S.
(b) (10 pts.) Let G(x, y, z) = 3y i − xz j + yz 2 k. Find ∇ × G and set up, but do not
evaluate, an iterated double integral that gives the surface integral
ZZ
∇ × G · n dS .
Σ
(c) (15 pts.) Use Stokes’s Theorem to convert the surface integral in part (a) into a line
integral, and then evaluate the line integral.