A few problems and refferences on Multivariable Calculus

September 16, 2004

1. (a) Let f ∈ C^{1}(ℝ^{n}, ℝ^{n}) be such that |Df_{0}| = 0. Show that if a local inverse of f

exists, it can not be differentiable.

(b) Let f ∈ C^{1}(ℝ^{n}, ℝ^{n}) be such that ∀x ∈ ℝ^{n}, |Df_{x}| ≠ 0. The inverse function theorem guarentees that ∀x ∈ ℝ^{n} a local inverse of f exists. Show that a global inverse need not exist.

2. (a) Suppose f ∈ C^{k}(ℝ^{n}, ℝ^{n}) and |Df_{0}| ≠ 0. Show that f^{−1} is also C^{k}. Hence if

f ∈ C^{∞} then f^{−1} ∈ C^{∞}.

(b) If f : ℝ^{n} → ℝ^{n} is real analytic, and |Df_{0}| ≠ 0 show that f^{−1} is also real analytic.

3. Let f ∈ C^{k}(ℝ^{m+n}, ℝ^{m}) with f(0) = 0. If det(D_{0}f|_{ℝ^{m}×0}) ≠ 0 the implicit function theorem tells us that ∃U ∋ 0 and a unique function g such that ∀y ∈ U, f(g(y), y) = 0. Show that g is C^{k}. What happens when we replace C^{k} with real analytic?

4. We say f : ℝ → ℝ^{2} is a space filling curve if f is continuous and surjective. A lucid proof of the existence of such curves can be found in Munkres.

(a) Show that no space filling curve can be C^{1}. Can it be differentiable?

(b) (Unrelated) Show that no space filling curve can be injective.

(c) (Unrelated) Suppose f ∈ C^{1}(ℝ^{2}, ℝ). Show that f is not injective.

5. Let U and V be open subsets of ℝ^{m} and ℝ^{n} respecitvely. If there exists a C^{1} bijective function f : U → V show that m = n. (Hence ℝ^{m} is not diffeomorphic to ℝ^{n}).

References

[1] Spivack, Calculus on manifolds [nice proofs of the inverse / implicit function theorem, and a solid foundation of multivariable calculus]

[2] Apostol, Mathematical Analysis

[3] Milnor, Topology from a differentiable point of view [Nice proof of Sards theorem, and an introduction to degree theory]

[4] Guilemen and Pollak, Differential Topology [Good for the soul]

[5] I’m sure the first Rudin does this junk, but I never read it.