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) | | | | | | |
|