Honors Exam in Real Analysis 2010
1. In this problem, all matrices and vectors are given with respect to the standard basis.
a) Find the dimension of A2(R4), the space of alternating 2-tensors on R4.
b) Verify that the following map f: (R4)2 → R is a member of A2(R4).
c) Compute
where f is as above and T: R2 → R4 is the linear transformation given by the matrix
2. Let U ⊂ Rn-1 be an open set, and let α: U → Rn be a smooth coordinate patch (in particular, α has Jacobian of rank n - 1 everywhere). Let g: Rn → R be a smooth function such that g(y) = 0 and Dg(y) ≠ 0 for all y ∈ α(U).
a) Let f: Rn → R be a smooth function, and supppose that the restriction of f to α(U) has a strict maximum: that is, there is some p ∈ α(U) such that
f(p) > f(q) for all q ∈ α(U) \ {p}.
Prove that Df(p) = λDg(p) for some real number λ.
b) Use part a) to find the maximum of the function f(x, y, z) = x + y + z on the set α(U), where U = {(x, y) : 0 < x < 1 and 0 < y < 1 }
and
α(x, y) = (x, y, 6x - 6x2 + 3y - 3y2).
(Hint: Take g(z, x, y) = z - 6x + 6x2 - 3y + 3y2.)
3. In multivariable calculus, we learn about integration with polar coordinates. Here is a particular case: Suppose that U is an open bounded region in the x-y plane that is easily described in polar coordinates by a ≤ r ≤ b and c ≤ θ ≤ d - that is,
U = {(r cos(θ), r sin(θ)), r ∈ (a, b) and θ ∈ (c, d)},
where (a, b) ⊂ (0, ∞) and (c, d) ⊂ (0, 2π). Suppose that f: U → R is a smooth function. Then we have that
∫Uf(x, y) = c∫d a∫b f(r cos(θ), r sin(θ))r dr dθ.
Give a rigorous justification for this formula.
-4. Let S2 be the unit sphere in R3. Compute
∫S^2 xz dx ∧ dy + 3y dx ∧ dz + y dy ∧ dz.
5. Let [a, b] be an interval in R, and let α : [a, b] → R2 be a smooth coordinate patch (i.e. α is 1-to-1, smooth on (a, b), has nonzero Jacobian everywhere on (a, b), and has continuous inverse on its image).
Let us write γ = α((a, b)) for the image of α in R2.
In advanced calculus we define the length of γ in two ways. One is the "surface measure" formula
∫γ1 dS = a∫b det M(t),
where M(t) is the 2 × 2 matrix whose first column is Dα(t) (the Jacobian of α at t) and whose second column is the unit normal to γ at α(t), chosen so that det(M(t)) > 0.
The more common definition of the length of γ (as given, say, in Munkres' Analysis on Manifolds) is the "one-dimensional volume" formula
∫γ1 dV = a∫b V (Dα(t)),
where
V(D(α(t)) = √(det [Dα(t)trDα(t)].)
a) Show that these two definitions of the "length of γ" actually agree.
b) Rewrite the integral ∫γ 1 dS as the integral of a 1-form on γ.