Math 3220
Introduction to Analysis
Summer term, 2000

Homework Assignments

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Homework due Monday July 31

From the text:
   Section 11.5 (second edition): #1ac, 6, 2ac, 3
     (=page 318-320 first edition, #6 new = #2 old, #3 new = #5 old. Numbers 1ac, 2ac are written below.)
   Section 11.6 #1, (=page 328 #1 first edition). THIS PROBLEM IS NOW OPTIONAL, AS IS ALL OF SECTION 11.6. SOLUTIONS WILL BE PROVIDED, AS WELL AS OPTIONAL CLASS NOTES FOR 11.6.


HW1: Recall an affine map f:R^n --> R^m is given by f(x) = Ax + b, where A is an m by n matrix, x is an n-vector, and b an m-vector. Prove that for such an affine map, one has the approximation formula

     f(x+h) = f(x) + Ah + E(h),

where in fact the error E(h) is identically zero. Deduce that f is differentiable, with derivative matrix A.

HW2: Prove that an affine map f(x) = Ax + b from R^n --> R^n has an inverse function if and only if the matrix A is non-singular, and that the derivative of the inverse map is A^(-1). (Hint: if y=Ax+b try to solve for x in terms of y to get the inverse map.)


For those of you with old editions, here is the text from 1ac, 2ac above:

1) For each of the following functions, prove that f^(-1) exists and is differentiable in some nonempty, open set containing (a,b) and compute D(f^(-1))(a,b).
(a) f(u,v)=(3u-v, 2u+5v), at any (a,b).
(c) f(u,v)=(uv,u^2 + v^2), at (a,b)=(2,5). (Note: this point corresponds to four possible values of (u,v), i.e. (2,1),(1,2),(-1,-2),(-2,-1), so there are four possible ``branches'' of the inverse function which you may choose, and you will get different answers for your corresponding derivative matrices depending on which branch you pick.

2) For each of the following functions, find out whether the given expression can be solved for z in a nonempty open set V containing (0,0,0). Is the solution differentiable near (0,0)?
(a) xyz + sin(x+y+z) = 0.
(c) xyz(2cos(y) - cos(z)) + (zcos(x) - xcos(y))=0.