Math 3220
Introduction to Analysis
Summer term, 2000

Homework Assignments

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or to our grader/tutor Darrell Poore

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Homework due July 14

From the text:
   Section 11.3 page 333 (second edition), #5,7,13.
   Section 7.4 page 217 (second edition), #8; (= #8 section 4.7 page 204 in first edition).

Also, from the notes of Friday July 7:
   HW1: Prove that if f:R^n --> R^m is differentiable at a, then f is continuous at a.
   HW2: Prove that if f is continuous at a, then there exists positive r and finite M so that ||f(x)|| is less than or equal to M for all x in the ball of radius r centered at a.
   HW3: Consider the polar coordinate map (x,y)=f(r,theta)=(rcos(theta), rsin(theta)), and its inverse map (r, theta)=g(x,y)=(sqrt(x^2 + y^2), arctan(y/x)), (on suitable subdomains of R^2).
  (a) Show that the partial derivative of r with respect to x does not equal the reciprocal of the partial derivative of x with respect to r.
   (b) Find an actual formula for partial r with respect to x in terms of the four partial derivatives of x and y with respect to r and theta. (In this case you may end up with a formula which wouldn't work for the general case of an arbitrary invertible map from R^2 to R^2, but see if you can derive such a general formula using the adjoint formula for the inverse of a matrix.)

Also, from class on Monday July 10:
   HW4: Use Taylor's Theorem to find an approximate value for the following quantities, with and error less than 10^(-6).
     (a) sqrt(4.1)
     (b) the natural logarithm of 1.2
   HW5: This is the problem from 7.4 listed above.

Grading: 22 points possible, distributed as follows: 11.3 #5a: 1 point; 5b: 2 points; 7a: 2 points, 7b: 1 point; 13: 2 points; HW1: 2 points; HW2: 2 points; HW 3a: 1 point; HW 3b: 1 point; HW4a: 2 points, 4b: 2 points; HW5 (7.4#8) a: 2 points, b: 2 points;