MATH 3033 Real Analysis
Homework 3
Due on April 14, 2022
1. Let f : R3 → R2 and g : R2 → R3 be defined by
f(x, y, z) = (2x3 − 3z − 1, x− 5y + xz)
g(u, v) = (uv2, 2v + u, u2 + v)
(a) Find Df(x, y, z) and Dg(u, v).
(b) Compute D(f ◦ g)(1, 1).
2. Let F (x, y) = y2 + y + 3x + 1 = 0
(a) Find the set of x for which we can solve the equation for y in terms of x without using implicit
function theorem.
(b) Now do the same using implicit function theorem. Do the results agree?
(c) Compute
dy
dx
.
3. Show that the system of equations  u = x + xyzv = y + xy
w = z + 2x + 3z2
is solvable for (x, y, z) near (u, v, w) = (0, 0, 0).
4. Consider F (x, y, u, v) = (z, w) such that
z = x2 − y2 + u2 − v3 + 4
w = 2xy + y2 + 3u4 − 2v2 + 8
(a) Show that F (2,−1, 1, 2) = (0, 0) and prove that there exists open sets U and V in R2 contain-
ing (2,−1) and (1, 2) respectively, a continuously differentiable function g : U → V such that
F (x, y, g(x, y)) = (0, 0) for all (x, y) ∈ U .
(b) Suppose g(x, y) = (g1(x, y), g2(x, y)), where g is the function in (a). Write down explicitly
∂g1
∂x
and
∂g2
∂y
.
5. For n ∈ N, let fn(x) = e
−n2x2
n
for x ∈ R.
(a) Prove that fn → 0 uniformly on R.
(b) Prove that f ′n → 0 pointwise on R but the convergence of {f ′n} is not uniform on any interval
containing 0.
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