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    Problem Sets 1-4 include references to the textbook: Apostol, Calculus, Vol. I, Second Edition (1967), and Problem Sets 5-10 include references to Vol. II of the book.

    PROBLEM SET # PROBLEMS SOLUTIONS
    1

    Due in Ses #6 in class

    1. Ex. 25 on pp. 457 and Ex. 15 on pp. 468.
    2. Ex. 4 on pp. 613.
    3. Prove that if W is a nontrivial subspace of V_n, then W has an orthonormal basis. (Hint: induction on dimension of W)
    		 Solution 1 (PDF)
    2

    Due in Ses #8 (before the class)

    1. Ex. 14, pp. 483.
    2. Ex. 12, pp. 604.
    3. Find the reduced echelon form of a square matrix of whos rows form a basis in V_n. Prove your answer.
    		 Solution 2 (PDF)
    3

    Due in Ses #11 (before the class)

    1. Show that 3 planes in V_3 intersect in a single point if and only if their normal vectors are linearly independent.
    2. Using Gauss-Jordan reduction and the properties of det, compute the determinant and the inverse of the 4x4 matrix:
      2 -1 0 0
      -1 2 -1 0
      0 -1 2 -1
      0 0 -1 2
    3. Ex. 3 on pp. 492.
    		 Solution 3 (PDF)
    4

    Due in Ses #17

    1. Ex. 14, pp. 529.
    2. Ex. 11, pp. 539.
    3. Consider the curve given in polar coordinates r and \theta by the equation r= exp(-\theta), where \theta changes between 0 and 2\pi M for a positive integer M. Find the length of this curve. What happens as M becomes arbitrarily large?
    		 Solution 4 (PDF)
    5

    Due in Ses #20

    1. Ex. 22, pp. 256.
    2. Ex. 8, p.p 263.
    3. Ex. 10, pp. 269 (Hint: there is a one-line solution).
    		 Solution 5 (PDF)
    6

    Due in Ses #23

    1. Ex. 14, pp. 276.
    2. A rectangular box has volume 10 cubic inches. Find the dimensions that minimize surface area.
    3. Ex. 18, pp. 313.
    		 Solution 6 (PDF)
    7

    Due in Ses #26

    1. Ex. 13, pp. 332.
    2. Ex. 8, pp. 337.
    3. Ex. 16, pp. 345.
    		 Solution 7 (PDF - 1.3 MB)
    8

    Due in Ses #29

    1. Show that if the double integral of f(x)=1 over a bounded set S exists, then the boundary of S has content zero. (Hint: use the Riemann criterion).
    2. Ex. 7 on pp. 371.
    3. Let f(x,y)=1 if {x =1/2 and y is rational}, and f(x,y)=0 otherwise. Show that the double integral of f(x,y) over Q=[0,1]x[0,1] exists, but the ordinary integral of f(x,y) dy from 0 to 1 fails to exist when x =1/2.
    		 Solution 8 (PDF)
    9

    Due in Ses #34

    1. Ex. 4 on pp. 391.
    2. Let R be a Green's region on the plane with boundary C, and suppose that f(x,y)=f_1(x,y)i +f_2(x,y)j is a continuously differentiable function on an open set containing R. If S is the length function of the curve C, and n is the outer normal of C, express the path integral of the dot product (f.n)dS along the closed curve C as a double integral over R.
    3. Ex. 8 on pp. 386.
    		 Solution 9 (PDF)
    10

    Last problem set due in Ses #37

    1. Ex. 18 on pp. 400.
    2. Ex. 20 on pp. 415.
    3. Ex. 10 on pp. 430.
    		 Solution 10 (PDF - 1.3 MB)