Math 444, Section E12, Spring 2008

Math 241 F1H, Honors version, Fall 2008

Homework assignments Score Reports Qz1 (16.5/20) Qz2 (17/20) Ex1 (83/100) Ex2 (82.2/100)Qz4(14.4/20) Qz5(15.6/20) Qz6(16.7/20) Ex3(82.9/100) Final (85/100 with extra points counted)

. Lectures: MTWTh 2:00-2:50 in 142 Henry Bld

. Instructor: Tao Mei, mei@math.uiuc.edu

. Course homepage: www.math.uiuc.edu/~mei/math241.html

. Office & Telephone: ALTGELD Hall 109, 217-244-2478

. Office Hours: 5:00-6:00pm Monday and Wednesday or by appointment.

. Textbook (required): Calculus, 3^rd edition: (written by) R. Smith, R, Minton, ISBN: 978-0-07-287029-9.

Course Description: The focus of this course is vector calculus, which concerns functions of several variables and functions whose values are vectors rather than just numbers. In this broader context, we will revisit notions like continuity, derivatives, and integrals, as well as their applications (like finding minimal and maxima). We'll explore new geometric objects such as vector fields, curves, and surfaces in 3-dimension-spaces and study how these relate to differentiation and integration. The highlight of the course will be theorems of Green, Stokes, and Gauss, which relate seemingly disparate types of integrals in surprising ways.

Differences to the standard Math 241: The course covers the same material as the regular 241 sections, but much deeper. It is more challenging and more labor-intensive, but also intellectually more rewarding than the standard version of 241. For example, we will cover theoretical material that in standard sections would often be skipped, we will do some proofs, and the homework assignments will include more difficult, "honors level" (but doable) problems. This course qualifies for honors credit for students in the James Scholar or Campus Honors programs. It is taught in small sections in four lecture/discussion hours per week, instead of the standard format of three hours of lectures (in large classes) and two hours of recitation (in small classes, run by TA's). The only way to get in the class is by registering online IF a space opens up by the second Tuesday. Send email to advising@math.uiuc.edu for registration questions.

Tentative exam dates: There will be three hour-long exams during the semester and a final exam. The class on the following Thursday will be dropped.

Exam #1 -7:00-8:00pm, Wednesday, Sep. 24^th in Altgeld 243.

Exam1 Covers Chapter 10-11. Besides quiz and hw problems, you can try the following problems from textbook to get ready:

Page 917: #15, 27, 32,36,37,39, 44-46. Page 850: # 3,7,27,29,31,41,43,55,57.

Answer: #32: <(t^3)/6+2t+4,(e^{2t})/4-t/2-1/4>, #36:56 feet,100+40(sqrt7) feet, 82 ft/s, #42 k(0)=infty, k(2)=29/(592*sqrt74), #44 <-1,0,0>, <0,-1/sqrt2, -1/sqrt2), #46: a_T(0)=0, a_N(0)=2, a_T(2)=4/sqrt5, a_N(2)=2/sqrt5.

Exam #2 -7:00-8:00pm, Wednesday, Oct. 22^th in Altgeld 243.

Exam2 covers Chapter 12 and the first 3 sections of Chapter13. Besides quizzes and hws, try the following problems of the textbook:

Page 1026: #17, 19, 31,37, 41, 45,47, 51, 55, 61, 65, 67. Page 1111 # 3, 23, 27, 29.

Exam #3 -7:00-8:00pm, Wednesday, Nov. 19^th in Altgeld 243.

Exam3 covers through section13.4 to section 14.4. Besides quizzes and hws, try the following problems of the textbook:

Page 1111 # 37,45,46,51.53,56,58,69,71, Page 1217 #5, 9, 17, 21, 25, 27, 31, 35.

Answer for even number questions: #46, 0; #56, (a)r=3 (b) \ro\sin\phi=3; #58, (a) \theta=\pi/4, 5\pi/4 (\ro=0 is included already); (b) \theta=\pi/4, 5\pi/4 (\phi=0,\pi is included already).

Final Exam- 1:30-4:30 PM, Tuesday, December 16 in our regular classroom (142 Henry building). Please arrange any travel, etc., so that you can take the final on this date. As a "non-combined" course, there will be no conflict exam given except for those few individuals who meet the official university criteria given here in the student code.

Review probs for sections 14.5-15.8, page 1220, #63, 69, 71,75, 77.

Homework and Quiz: Practice is essential for this course. I will assign a few homework problems after each class. You should check the assignment at the class webpage. Problems with a “*” are required to be handed in weekly. Handed-in homework is due at the beginning of class on the due date. Assignments dropped off in mailboxes will not be accepted; however, you can turn in an assignment, in person in my office any time before the class hour in which it is due. We will also have practicing quizzes during the week before/of the exams. 3 poorest grades of Hws and quizzes will be dropped from the final score in the class. As a result HW extension or make-up quizzes will not be allowed.

Homework Rules: (1) Print your name, section at the top of the paper. (2) Staple the papers together if your homework takes more than one page. (3) Work out the problems, grading will be based not only on the correctness of the solution, but also on the quality of the write-up and the work shown; in particular, an answer alone (like the answers to odd-numbered problems given in the back of the book), without justification, won't earn credit.. (4)Be neat and legible.

You will be able to check your homework and exam grades at Score Reports, which is the Math Department's gradebook program. This will be available beginning a few weeks into the semester. All your hw, quiz and exam grades will be posted there. You will be prompted for your NetID and NetID password after following the Score Reports link. Please check score reports regularly to make sure your grades have been correctly reported and tell me promptly about any errors. You are responsible for keeping all of your graded homework and exams so that any discrepancies in recorded grades can be settled.

Grading: three midterm (15% each), final exam (30%), Home-works & Quizzes (25%).

How to succeed in this class

Remember this is an honors level course, aimed at the best and brightest of our students. As such, it is more challenging and labor-intensive as a regular calculus course, and it requires a significant level of commitment and work - probably more so than any of your other classes. To do well in this class, you must:

Attend the lectures.
Read the text.

Do the reading right before/after the lecture (or the same day), when the lecture is still fresh in your mind. Don't wait with the reading till the homework is due. A 5-minutes review of old text and pre-reading of the new text will be very helpful.
Read each section in linear order, from beginning to end, as you would read a novel. The material in the text is usually arranged in a logical order, with each concept building naturally on the previous one. You get the most out of it if you read the section in this order. Resist the temptation to treat the text as a dictionary/encyclopedia, looking up concepts and formulas only when you need them in a homework problem!
Don't neglect the pictures. A picture can often explain a concept better than a passage of prose.
While the bulk of the reading should be done after I have covered the section in class, it is useful to read, or at least skim, the text of the upcoming section ahead of each lecture. This will help you in understanding the lecture, and it may uncover any difficult points in the material that I should go over in the lecture.

Take the homework seriously. The homework, both the non-graded and the graded variety, is an essential part of this course, and requires a substantial investment of time. If you skip the non-graded homework, or blindly copy someone else's homework without really understanding what's going on, you'll be in trouble during exams.
Come to office hours and get help from your classmates.

Homework Assignments (A complete assignment ends with a .. An assignment end with a ; is incomplete, additional problems will be added to it later. Only problems with a “*” are hand-in problems. )

Course hotline (email to students and instructor): Share your questions on homework, course material…

Assignment #1 (due by the beginning of the class on Sep. 2^nd, Tuesday)

(a) Read: Example 1.2, 1.3, the proof of Th1.1, Th3.1, 3.4, Example 3.6,

(b) Ex10.1: #1, 5, 25, 35, 38, 52, Ex10.2: #11, 13, 21, 23, 27, Ex 10.3: #3, 5; Ex10.4, #5,

(c) Ex 10.3: #9*, 13*, 21*, 29* (read example 3.6 first), 34*, 43*(assume Th3.3 is true for vectors in n-dimension space), 48*, 49*, Ex 10.4: #12*, 18*, 64*.

Hint for Ex10.3, #43: Try the case n=3 first by applying Theorem 3.3 to vectors a=<p1, p2, p3>, b=<1, 1, 1>. Then prove the general case by assuming Theorem 3.3 is true for n-dimension vectors.

Grade average 17.2/20, (#29, 34, 43, 48 were not graded, #64=5pts, #18,21,49=9pts, other 3 pbs =6pts)

Assignment #2 (due by the beginning of the class on Sep. 9th, Tuesday:

(a) Read: Def 4.2, Example 4.7, 5.8, Example 6.6-6.8, Finish the calculations I skipped in the proof of Theorem 4.4.

(b) Ex10.4: #19, 41-46, 56, 62( coplanar means all vectors are in the same plane), 65, 67, Ex 10.5: #2,6, 17,22 , 55-62, 67. Ex10.6:#12,17, 22, 47, 52.

(c) Ex10.4: #24*,50*, 52*, Ex10.5: #41*,48*, 54* , Review Ex(page850): 50*,58*, Find the parametric equation of a line lying on the surface x^2+y^2-z^2=1 and passing through the point P=(1,-1,1).* How many such lines we have?*

Hint for the last problem: Let us assume we get such a line L and assume L is parallel to a vector a=<a1, a2, a3>. Write down the parametric equation of L assuming the vector a is known. Then find <a1, a2, a3> by the fact that L is on the surface x^2+y^2-z^2=1 (this means ALL (x,y,z) on L satisfies x^2+y^2-z^2=1). Grade average: 17/20, (#24, 52, 41, 48=12pts, #54,50=8pts).

Assignment #3 (due by the beginning of the class on Sep. 16th, TuesdayJ

(a) Read: Example 1.5, Example 3.3, page 880-882 on equation of motion, Page 892 the paragraph containing equality (4.8).

(b) Ex11.1: #1,2,12,29,31,44, Ex11.2#5,7,16,21,26,34,42,51,53, Ex11.3:#8,13,33, Ex11.4# 6, 49.

1newton/kg=1meter/s^2=3.28feets/s^2 (I used it incorrectly in class). Hint: #42, A change of variables will show that the two integrals are the same.

Grade average=15.56/20, Ex11.1#42=3pts #47=2pts 11.2#45=3pts #47=3pts 11.3#43=5pts 11.4#16=2pts #19=2pts

Assignment #4 (due by the beginning of the class on Sep. 23th, TuesdayJ

(a) Read: Example 5.1,5.4, page 903-906 about Kapler’s law.

(b) Ex11.5: #4,9, 19,20, 21,25-28, 30,33, 39, Ex11.6: #7,21,41-43,46.

(c) Ex11.5:# 16*, 18*, 29*, 31*, 34*, Ex11.6#44*, Let r(t) be a vector valued function, prove that if r’’(t) is parallel to r(t) for all t, then the curve defined by r(t) lies in a plane containing the origin*.

Grade average=15.6, #16 4pts #18 2pts #29 3pts #31 3pts #34 2pts #44 3pts.

Assignment #5 (due by the beginning of the class on Sept. 30th, TuesdayJ

(a) Read: Example 2.4, 2.5.

(b) Ex12.1# 4,8, 11,15,37,47,48,49, Ex12.2#1, 3, 7, 10.

Grade average=18.7/20, 4pts for each prob.

Assignment #6 (due by the beginning of the class on Oct. 7th, TuesdayJ)

(a)Read: Def 2.3-2.4, Example 2.8, 2.10, 2.11, Example 4.5, Example 5.2, Example 5.5. 5.6.

(b)Ex12.2# 5,19, 25,51,52,54, Ex12.3#3,5,9,13,29,33,45,48,53,62,63, Ex 12.4 #1, 7,23, Ex 12.5 #3, 17, 21, 25-27.

Hint for #61: Once you find f_x, use the definition (f_x)_y(0,0)=lim (f_x(0,h)-f_x(0,0))/h to find (f_x)_y at (0,0). Do not try to differentiate f_x in the y direction, that will be really complicated and does not provide the value of (f_x)_y at (0,0).

Grade average=15.7/20, 12.3 #61=2pts 12.4 #2 #8 #24=3pts each 12.5 #4 2pts #20 3pts #22 4pts.

Assignment #7 (due by the beginning of the class on Oct. 14th, TuesdayJ)

(a)Read: Example 6.5, 7.4, 7.5, Proof of Theorem 7.2 on page 1007.

(b)Ex 12.6 #3, 13, 29,37, 41, 47-52, 68, Ex12.7 #6,7,51-54, 57-60, Ex12.8, #3,11,17,33,43, 47.

(c) Ex 12.6 #4*, 14*, 36*, 42*, Ex 12.7, #4*, 38*, 49*, Ex12.8 #12*,18*, 48*, Let f(x,y)=x^3-2(y^2)+3(x^2)y, show that (0,0) is a critical point and D(0,0)=0. Find whether (0,0) is a local max , local min or a saddle point*.

Grade average=16.5/20, #42=2pts, #38=4pts, #49=4pts, #18= 3pts, #48=3pts, web question=4pts.

Assignment #8 (due by the beginning of the class on Oct. 21, TuesdayJ)

(a)Read: Example 2.1, 2.5, 3.5.

(b) Ex 13.1 #7,9, 15,17,43,54,57,31,7, Ex13.2 #3, 43, Ex13.3 #5,6, 29, 39.

Assignment #9 (due by the beginning of the class on Nov. 4, TuesdayJ)

(a)Read: Example 5.5, Example 6.2, 6.3.

(b) Ex 13.4 #5, Ex 13.5 #5, 23, 43, 44, 45, 49, Ex 13.6 #3,7,13,19, Ex13.7 #3,11,23,27.

(c) Ex 13.4 #6*, Ex 13.5 #10*, 28*, 46*, Ex 13.6 #22*, 31*, 36*, Ex 13.7 #36*, 42*, 54*, Consider the portion of the unit sphere between two parallel planes, prove that its surface area only depends on the distance between the two planes*. (hint: Since the unit sphere is symmetric with respect to the origin, it is enough to show the desired result for the case that the two parallel planes are z=a, z=b with -1<a<b<1).

Assignment #10 (due by the beginning of the class on Nov. 11, TuesdayJ)

(a) Read Example 1.4, 1.10, 2.3.

(b) Ex13.8 #9, 11, 23, 32, Ex 14.1 #11, 45-48, Ex 14.2 #5.

(c) Ex13.8 #30*, 33* (assume #34), Ex14.1 #28*, 30*, 38*, Ex 14.2 #8*, 52*, Let C be a curve traced out by <x(t), y(t)> with 0<t<1. Suppose u(t)=x(-t), v(t)=y(-t), then the curve C’ traced out by <u(t),v(t)>, -1<t<0 is the same as C but with opposite orientation. As time t increase, C is from (x(0),y(0)) to (x(1), y(1)) while C’ is from (x(1),y(1)) to (x(0),y(0)). Prove that the line integral of f(x, y) with respect to arc length along C and C’ are the same.* (hint, use Th2.1 and change variable).

Assignment #11 (due by the beginning of the class on Nov. 18, TuesdayJ)

(a) Read Example 2.9, Theorem 3.4.

(b) Ex 14.2 #7, 27, 51, Ex 14.3 #15, 21, 37,45, Ex 14.4, #3, #11, #25.

Assignment #12 (due by the beginning of the class on Dec.9, TuesdayJ)

(a) Read page 1180-1182 on parametric representation of surfaces, Example 6.7, proof of Theorem 7.1, 8.1.

(b) Ex 14.5 #3, 11, 15, 27, 31-36, Ex 14.6 #7, 21, 33, 45, Ex 14.7 #3, 9, 27, Ex 14.8, #3,7, 19,27.

(c) Ex 14.5 #28*, 44*, Ex 14.6 #34*, 46*,68* (choose either an outward or an inward normal vector), Ex 14.7 #10*, 28*, Ex 14.8 #10*, 20*, 29*.

Assignment #13 (no due date): Read section 14.9 and prepare the final.