AMCS/MATH 608 Problem set 0, September, 2014 Dr

AMCS/MATH 608
Problem set 0, September, 2014
Dr. Epstein
Reading: References for this material are Principles of Mathematical Analysis, by Walter Rudin, The Way of Analysis by Robert Strichartz, Elementary Classical Analysis, by
Jerrold Marsden, and Calculus on Manifolds, by Michael Spivak.
Recall that an Rm -valued function, f, defined in an open set U ⊂ Rn is differentiable
at x0 ∈ U, if there is a linear transformation A : Rn → Rm so that
f (x) = f (x0 ) + A(x − x0 ) + o(kx − x0 k).
(1)
We use D f (x0 ) to denote A. The function is differentiable in U if it is differentiable at
every x ∈ U, and continuously differentiable if the map x 7→ D f (x) is a continuous map
from U to m × n matrices.
The function f is twice differentiable if the map x 7→ D f (x) is differentiable, and
2
continuously differentiable if x 7→ D(D f )(x) is continuous as a map from U to Rmn .
You should think about this so you understand how to interpret D(D f )(x) as a family of
quadratic forms with values in Rm .
These problems, which do not have to be handed in, cover material in advanced calculus and exterior calculus. You are expected to know this material and are strongly encouraged to do these problems carefully.
Standard Problem
1. Suppose that f (x, y), defined in [0, 1] × [0, 1], has partial derivatives in (0, 1) ×
(0, 1). If ∂ y f (x, y) = 0 throughout (0, 1)×(0, 1), show that there is a differentiable
function g(x) so that f (x, y) = g(x). How is ∂x f (x, y) related to g 0 (x)?
2. Define f : R2 → R by
(
f (x, y) =
2
2
−y
x y xx 2 +y
2
0
if (x, y) 6= (0, 0)
if (x, y) = (0, 0).
(2)
(a) Show that f is differentiable in a neighborhood of (0, 0) with ∂x f (0, y) = −y
and ∂ y f (x, 0) = x.
(b) Show that ∂x ∂ y f (0, 0) and ∂ y ∂x f (0, 0) exist but are not equal.
1
(c) Why does this example not contradict the theorem that “mixed partials commute?”
3. Let f : R2 → R be continuously differentiable. Show that f is not one-to-one.
Hint: If, for example, ∂x f (x, y) is not zero, then show that the map F : (x, y) →
( f (x, y), y) is locally one-to-one and onto.
More challenging problems:
1. Let ω = f (x, y)d x + g(x, y)dy be a 1-form defined in U ⊂ R2 and H (s, t) : V →
U a C 2 -map. Show that
H ∗ (dω) = d(H ∗ ω).
(3)
Is it necessary for H to be 1-1?
2. Let γ : [0, 1] = R2 be an embedded C 1 -curve in the plane, and let p = { p0 , . . . , p N },
where pi = γ (i/N ). Finally, let γ p be the polygonal curve obtained by joining successive points, pi , pi+1 , by the segments of the straight lines they define. We orient
γ p so that the vertices come in the given order. We let
| p| = max{| pi+1 − pi | : i = 0, 1, . . . , N − 1},
and ω be a C 0 1-form defined in a neighborhood of γ . Prove that, as | p| → 0,
Z
Z
ω converges to
ω.
(4)
γ
p
3. Suppose that S1 is the unit circle in the plane, centered at (0, 0), which we approx2πi j
imate by a polygonal “stair case”-curve, γ N obtained as follows: let p j = e N =
(x j , y j ), j = 0, . . . , N . We join p j to p j+1 by the arc:
(
(1 − 2t)(x j , y j ) + 2t (x j , y j+1 ) for t ∈ [0, 21 ]
c(t) =
(5)
(2 − 2t)(x j , y j+1 ) + (2t − 1)(x j+1 , y j+1 ) for t ∈ [ 12 , 1].
Draw this curve for a few values of N . Show that
Z
Z
lim
ds N = 8 and lim
xdy = π,
N →∞
γN
N →∞
γN
where ds N is arclength along the curve γ N , and xdy is the 1-form.
2
(6)
4. Define a 1-form in R2 \ {0} by setting
xdy − yd x
x 2 + y2
ω=
(7)
(a) Prove that dω = 0, but show that there is no function f defined in R2 \ {0} so
that d f = ω. Hint: what is
I
ω?
(8)
x 2 +y 2 =1
(b) How about in the upper half plane {(x, y) : y > 0}?
(c) Show how to use the 1-form ω in Stokes theorem to compute
Z1
dt
.
1 − 2t + 2t 2
(9)
0
(d) Now let
η=
(x 2 − y 2 )d x + 2x ydy
.
(x 2 + y 2 )2
(10)
Show that dη = 0, and that there is a function g defined in R2 \ {0} so that
dg = η.
5. Suppose that U ⊂ Rn is an open convex set, and f : U → Rm is a continuously
differentiable function. Show that for x1 , x2 ∈ U,
f (x1 ) − f (x2 ) =
Z1
D f (t x1 + (1 − t)x2 )(x1 − x2 )dt.
(11)
0
Use this to show that in any compact subset K ⊂ U, and for any > 0, there is a
δ > 0 so that if x1 , x2 ∈ K and kx1 − x2 k < δ, then
k f (x1 ) − f (x2 ) − D f (x1 )(x1 − x2 )k ≤ kx1 − x2 k.
(12)
Show that if the operator norm, kD f (x)k < M for all x ∈ U, then
k f (x1 ) − f (x2 )k ≤ Mkx1 − x2 k.
3
(13)
6. Let U ⊂ Rn be an open set and f : U → Rn a continuously differentiable map,
such that, at any every point x ∈ U, det D f (x) 6= 0. Show that f (U ) is an open set.
7. Define a map from R2 to itself by setting
F(x, y) = (sin x cos y + sin y cos x, cos x cos y − sin x sin y).
(14)
Does there exist a point (x0 , y0 ) such that F is locally invertible in a neighborhood
of F(x0 , y0 ). You must prove your answer.
8. Let U ⊂ Rn and let f : U → R be a twice continuously differentiable function.
2
Suppose that at x0 ∈ U we know that ∇ f (x0 ) = 0 and the Hessian ∂x∂ ∂fx (x0 ) is an
j
k
invertible matrix. Show that for a δ > 0 there is a map G : Bδ (0) → U so that
G(0) = x0 and ∇ f (G(y)) = y. Show that there is a function g(y) defined in a
neighborhood of 0 so that G(y) = ∇g(y).
Finally suppose that U = R and there is a constant c > 0 so that f 00 (x) > c for all
x ∈ R. Show that G and g are defined on all of R as well.
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