Homework is to be collected in class on the dates indicated.

Homework #1: (due Friday February 20)

do Carmo Chapter ? problems ?

A. In class, we defined the notion of the differential of a map f:U-->R^m,
where U is an open subset of R^m, at a point p in U. This is a linear
transformation A:R^n-->R^m. Show that if the differential exists, it is unique.
Show that if n and m are both 1, then it coincides with the usual derivative,
interpreted as a linear map from R to R.  Show that linear transformations
are differentiable, and compute their differentials.

B. Let U and V be open subsets of R^n and R^m, respectively. Let f
be a map f:U-->R^m, and g be a map g:V-->R^k,  and let p be a point
of U such that f is differentiable at p, and such that f(p) is in V and
g is differentialbe at f(p). Show that the composition of f and g is
differentiable at p, with differential the composition of Df(p) and Dg(f(p)).

C. In class we defined for each i the map pi^i:R^n-->R taking
(x^1,x^2,...,x^n) to x^i. Show that this map is differentiable and compute
its differential. Similarly, fixing n-1 real numbers x^1, x^2,...,x^{j-1},x^{j+1},...,x^n
(omiting x^j), we can define a map

F^j_{x^1,x^2,...,x^{j-1},x^{j+1},...,x^n}: R-->R^n

taking x^j to (x^1,x^2,...,x^{j-1},x^j,x^{j+1},...,x^n). Show that this
map is differentiable and compute its differential.

D. The inner product on R^n can be thought of as a map i:R^{2n}-->R.
Show that this map is differentiable, and compute its differential.

E. Show that the set of invertible matrices is an open subset U
of L(R^n,R^n), with the metric space structure defined in class.

F. Show that the map f:U-->L(R^n,R^n) taking A to A^{-1}
is continuous.

G. Prove the implicit function theorem: Let U be a subset of
R^n x R^m, p be a point of U, and f:U-->R^n be differentiable everwhere.
Denote f(p)=q and p=(a,b) where a is in R^n,
and b is in R^m. Writing Df=[A B] with respect
to the standard basis of R^n, where A is an n x n matrix, and B an m x n matrix,
assume A(p) is invertible. Show that there exists an open subset U* of U,
containing p, such that U* intersect f^{-1}(q)={(g(u),u)} where
u ranges over some open set V* in R^m containing b, and g:V*-->R^n
is continuously differentiable and Dg=-A^{-1}B.

(NOTE CORRECTION IN THE SIGN!!!!)

H. Prove that if U is a subset of R^3 and f:U-->R is continuously differentiable,
and p is a point of U such that grad f(p) doesn't vanish, then denoting c=f(p),
show that f^{-1}(c) is locally the graph of a function over the xy, yz, or xz plane.

I. Let V be a vector field defined in a subset U, of R^n, i.e. V:U-->R^n,
and let p be a point of U. Assume V is locally Lipschitz at p, i.e., there exists a neighborhood,
B_s(p) and a constant C such that |V(q)-V(r)|<=C|q-r| for all
q and r in B_s(p).  Show that there exists a positive constant T>0, and
a continuously differentiable curve g:[-T,T] such that g(0)=p,
and Dg(t)=V(g(t)) for all -T<=t<=T. Show that if h is a differentiable
curve defined on an interval I containing 0, such that Dh(t)=V(h(t)) and h(0)=p, then
h coincides with g on [-T,T].

Homework #2: (due Friday February 27)

do Carmo Chapter 1 problems in sections
1-2 problems 1, 5,
1-3 problems 1-10,
1-4 problems 4, 6, 7, 10, 11
1-5 problems 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13

A. In class we showed

[ T '                  [ 0      k      0         [ T

N '        =       -k    0     -t            N

B ' ]                 0    t      0 ]            B ]

in the sense of matrix multiplication. Why is the square
matrix on the right hand side skew symmetric?

B. Show that the set of Euclidean motions defined in class
constitutes a group.

C. Show that if g:(0,l)-->R^3 is a C^3 curve with |g'|=1
and |g''|>0, with curvature k(s) and torsion t(s), then
the curve ~g given by composing g with a Euclidean motion
has the same curvature k(s) and torsion t(s)

D. Show that if the curvature function is C^1 and the torsion
is C^0 on the closed interval [0,l],
then it follows that the vector field on R^10 defined
in class is globally Lipschitz.

NOTE: D is incorrect as stated. One has to assume that the torsion is
more regular, C^1 will suffice. But nonetheless, the local existence theorem can be

modified to apply in this context. Why?
 

Homework #3: (due Friday March 12)

do Carmo Chapter 2 problems in sections
2-2 problems 1, 2, 3, 4, 8, 10, 12, 13, 16, 17
2-3 problems 1, 2, 3, 4, 7, 13
2-4 problems 2, 3, 4, 5

A. Check that the spherical coordinate parametrizations of the sphere endow
it with 2 local coordinate charts covering it.

B. Can the sphere be covered by a single coordinate chart?

C. Show that regular equi-sided and equi-angled n-polygons maximize area
over n-polygons of fixed perimeter. 

Homework #4: (due Monday March 29)

do Carmo Chapters 2-3 problems in sections
2-5 problems 1, 2, 3, 4, 5, 6, 9, 10, 11, 13, 14
2-6 problems 1, 5
3-2 problems 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 15, 17, 19, 20

Homework #5: (due Friday April 9)

Homework #6: (due Friday April 16)

Homework #7: (due Friday April 23)

Homework #8: (due Friday April 30)