M326 HW

Homework for Math 326, Winter 2014

General rules:

There will be weekly homework assignments due on Fridays. Hand in your assignment in class, or to the grader's mailbox, box 27 in PDL C-136, by 4 PM. Hand in whatever you can on time; some work is much better than nothing. Late homework usually will not be accepted; if you have exceptional circumstances that makes this policy a hardship for you, see me to discuss it.
Don't be stingy with space. Leave at least one blank line between problems, and leave margins on at least two and preferably all four edges of the page. If your writing shows through the paper, use only one side of the paper.
Label every problem clearly (page and/or section number as well as problem number).
Write your name, the class, the assignment number, and the due date in the upper right corner of at least the first page of your paper.
STAPLE your pages together. The "fold and tear" method of fastening pages is not acceptable.
Seven of the assignments will be collected and graded. Your total of homework points for the quarter will be rescaled to the appropriate number of course points: see the Grading Scheme for details.
Only some of the homework problems are graded for correctness. If a problem is not marked, you got credit one point for completing it, but your work was not checked. If there's a "0/1" mark for some problem, you didn't do enough work to get the 1 "completion point." (If the final answer or an answer with partial reasoning is in the back of the book, you won't get the completion point for merely copying what's in the back of the book.)

Assignments: The reading assignments indicate the sections of the text to read; the introduction for chapter n will be indicated by n.0. For example, §5.0 means pp. 116-117. The exercise numbers include the section.

Due:	Reading and Problems:
January 10	Read §§5.0-5.3. Exercises: p. 122, 5.1.2, 5.1.3, 5.1.4; p. 124, 5.2.1, 5.2.2; and p. 127, 5.3.3.
January 17	Read §§6.0, 6.2, & 6.4. (5.4 optional, should be review.) Exercises: p. 124, follow instructions of 5.2.3 for the function in 5.2.4(b). p. 124, 5.2.7 - you may use polar coordinates, and skip showing the inequality book asks for. Problem A. Let f(x,y) = (x³ - y³) / (x - y) if x ≠ y, and f(x,x) = 0. For each of the following points, determine whether f is continuous at the point, and justify your answer. (a) (0,0); (b) (2,0); (c) (2,2). CORRECTION p. 138, 6.2.3 (NOT 6.1.2); p. 153, 6.4.2. Problem B. Let f(x,y) = (x³ - 3xy²) / (x² + y²) everywhere except at the origin, where f(0,0) = 0. Compute f₁(0,0) and f₂(0,0). Then follow instructions of 6.4.7(c) (using the f defined here).
January 24	Read §§6.5, 6.1 & 6.6. Exercises: p. 127, 5.3.9; p. 153, 6.4.5, 6.4.7. Problem C, which is at the end of some Supplementary Notes on the Chain Rule (expanding on lecture from 1/17). pp. 160-161, 6.5.9, 6.5.15.
January 31	Midterm, no homework to hand in. Read §6.3; before Monday, Feb. 3, start reading §§6.7-8. Exercises (won't be collected, but do before the midterm): p. 134, 6.1.2 (Method II only, may use §6.6 formulas) & 6.1.3; p. 176, 6.6.1 (know how to get all the formulas, even if you don't write them all down) & 6.6.6; p. 138, 6.2.6; and pp. 143-144, 6.3.2 (Find all critical points in the region, and the maximum and minimum values and where they occur) & 6.3.14. Midterm information: general info and sample test; possibly more review problems later this week.
February 7	Read §§6.7-8 & 7.0-7.3. Exercises: pp. 161-162, 6.5.21, 6.5.22; p. 186, 6.8.1, 6.8.2, 6.8.10(a); and p. 199, 7.1.2. *NOTES on two problems: In 6.5.21, be sure your symbols r, h, and V don't look the same! In 7.1.2, show the partials exist everywhere, that they are not continuous at (0,0), and that f is not differentiable at (0,0). For the latter, show this in the same way we have been doing, ignore "To prove this ...".
February 14	Read §§7.4-7.6. RECOMMENDED: Before Monday, 2/10, review Taylor polynomials and series from a Math 126 text. (See also Thm. II, p. 99, but don't worry about formula (4.2-6) for the "remainder.") CHANGE: p. 201, 7.2.1 is postponed to a later homework assignment. This week's assignment: p. 210, 7.5.2 & 7.5.9; and the following two problems: Problem D. Let f(x,y) = (xy² - y³) / (x² + y²) everywhere except at the origin, and let f(0,0) = 0. (a) Find f₁(x,y) and f₂(x,y). (For each of the derivatives, you will need to give a formula that works everywhere except at the origin, and then the value of the derivative at the origin.) (b) Prove that f₁(x,y) and f₂(x,y) are not continuous at the origin. Are they continuous elsewhere? How do you know? (c) Prove that f(x,y) is not differentiable at the origin. Problem E: Let f(x,y) = x² - 4xy - 2y², and let R be the region x² + y² ≤ 20. (a) Find all critical points of f in the interior of R. (b) Use the method of Lagrange multipliers to find all candidates for extrema of f on the boundary of R. (c) What are the maximum and minimum values of f on R, and where do they occur?
February 21	Read §§8.0-8.3. Exercises: p. 201, 7.2.1; p. 220, 7.6.1(a)&(d) & 7.6.4; p. 229, 8.2.4 (Remember, "Justify your answer" is implied when a yes/no question is asked.) Problem F. Consider the portion of the solution set for x + y + z + sin(xyz) = 0 near the point (1,-1,0). Using the Implicit Function Theorem, what can we conclude about representing this as the graph of (a) x as a function of y and z? (b) y as a function of x and z? (c) z as a function of x and y?
Feb. 28	Midterm, no homework to hand in. No new reading this week, but before Monday, Mar. 3, start reading Chapter 9. Exercises (won't be collected, but do before the midterm): p. 177, 6.6.10: also determine conditions on a point P₀ = (x₀,y₀,z₀) needed to justify the assumption that there is such a differentiable function f. pp. 229-230, 8.2.3 & 8.2.20; p. 234, 8.3.1 & 8.3.2. Problem G is in the Regular points and the Implicit Function Theorem notes. For more information about the midterm, see the Test Information page and the Test Prep section of the class Study Materials workspace.
Mar. 7	Reading in Chapter 9: Start with Example 1, p. 239, and read the rest of the introduction. Read §§9.1, 9.2, and 9.3. In §9.4, read at least the one-page discussion of polar coordinates starting with equation (9.4-4), and also the lower half of p. 257 (noting the definition of a singular point for a coordinate system). Read §9.5. Exercises: In §9.2, use the following instructions. (a) Find the inverse of the transformation given. (b) In the xy-plane, draw several u-curves (that is, u = constant curves) and several v-curves. Also draw a separate uv-plane, with several x-curves and several y-curves. (c) Compute the Jacobian determinants of the two transformations (from (x,y) to (u,v), and the inverse), and show that they satisfy (9.1-6) on p. 242. p. 251, 9.2.2: Follow the instructions above, and draw R and R' in their respective planes, and compute their areas. p. 252, 9.2.6: Follow the instructions above. p. 252, 9.2.7: Follow the instructions above, and draw R and R' in their respective planes. Mark the boundary of R with arrows pointing counterclockwise; then find and indicate with arrows the corresponding motion around the boundary of R'. Then reread the last paragraph of §9.2. p. 262, 9.5.5: Follow the instructions in the problem, and also do (c) above.
Mar. 14	Finish Chapter 9 through §9.5 (see advice above) if you didn't already. Read §§15.32 & 15.62. In Chapter 15, don't worry about details of proofs, and read any other part of Ch. 15 you find useful. Exercises (not collected, but do before the final): p. 468, 15.32.3, 15.32.5, 15.32.9, & 15.32.11; pp. 497-498, 15.62.2(a) & 15.62.7. ALSO ON FRIDAY 3/14: Short, no notes, quiz on §§5.2, 5.3, 6.4, 7.1, & 7.2 which will count as part of the final exam. More information about this quiz will be posted by Monday.

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Most recently updated on March 6, 2014