Math 311-505 (Fall 2003)
Name
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Final Examination
Instructions: Show all work in your bluebook. Calculators that do linear
algebra or calculus are not allowed.
1. (10 pts.) Determine which, if any, of the sets below is a basis for P4 .
In any case, state why the set is or is not a basis.
S1 = {x + 1, 1 − x, x2 + x3 , 1 − x2 }
S2 = {1 − x − x2 , 1 + x, 1 − x2 , x3 , x3 − x2 }
S3 = {1 + x2 , 5 − 3x3 , 2 − x2 }.
2. Let L : P3 → P3 be given by L[p] := (1 − x2 )p′′ + (1 − 3x)p′ + 8p
(a) (5 pts.) Show that L is linear.
(b) (5 pts.) Find the matrix of L relative to the basis E = {1, x, x2 }.
(c) (5 pts.) Use part 2b above to find the the matrix of L relative
to the basis F = {x − x2 , x + x2 , 1 + x}. (Leave your answer as
a product of matrices.)
3. (10 pts.) Let x(t) = [x1 (t), x2 (t)]T , where x1 and x2 are the equilibribrium displacements for the masses in the spring system below.
Newton’s
law and
Hooke’s law yield the equation mẍ = kAx, where
−5 4
. Find the normal modes for the system.
A=
4 −5
k
m
4k
m
k
4. (10 pts.) Let (x, y) = T (u, v) = (u, veu ), so x = u and y = veu . If D∗
is the rectangle 0 ≤ u ≤ 1 and 0 ≤
Z v ≤ 1, then use Jacobi’s Theorem
e−x dxdy .
to change variables and compute
D
5. (10 pts.) A surface S is given parameterically by
x(r, θ) = r cos θi + r sin θj + (4 − r2 )k , 0 ≤ r ≤ 2, 0 ≤ θ ≤ π/4.
Find the standard normal, unit normal, both the vector and scalar
surface area elements, and the surface area of S.
2
2
y
x
6. (10 pts.) Let A be the area of interior
H of the ellipse a2 + b2 = 1. Use
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Green’s Theorem to show that A = 2 C (−ydx + xdy), where C is the
ellipse traversed once in the counterclockwise direction. Compute the
line integral to find A.
7. (15 pts.) Verify Stokes’s Theorem in the case where the vector field
is F(x) = (3y + 4x)i + (4x − y 2 )j + ez k, the surface S is the hemisphere
x2 + y 2 + z 2 = 9, z ≥ 0, with upward pointing normal, and C is its
properly oriented boundary, the circle x2 + y 2 = 9, z = 0.
Rπ
8. (10 pts.) Consider the real inner product hf, gi ≡ 0 f (x)g(x)dx
defined on C[0, π]. For the functions f (x) = sin(x) and g(x) = cos(x),
find the following: (a) kf k; (b) kgk; (c) the angle between f and g.
9. (10 pts.) Do one of these.
(a) Let V(x, t) be the velocity field of a fluid in three dimensional
space at time t, and let ρ(x, t) be the density of the fluid. There
are no sources or sinks for the fluid. Use Gauss’s Theorem to
derive the equation of continuity for the fluid, ∂ρ
∂t + ∇ · (ρV) = 0.
(b) State and prove Schwarz’s inequality for an inner product h , i
defined on an inner product space V .
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