ANNOUNCEMENTS
MTH 255/MTH 255H - Winter 2000

3/17/00

Below are the answers to the final; an answer key has been posted outside my office.
(Problem 8c was treated as extra credit.)
1. 3/25
2. (a) 2/3 i+1/3 j+2/3 k (b) 0
3. (a) conservative; g = xyz+xy+x+z2+C (b) not conservative
4. -9 Pi
5. 16/9
6. 16 Pi
7. -128 Pi/3
8. (a) 2 Pi; (b) 0; (c) 2 Pi     ((a) Just do integral; (b) Stokes' Thm; (c) Stokes Thm implies same as (a).)

3/13/00

If you wish to verify that the grades I have recorded are correct, either stop by my office today or tomorrow (mornings best), or check the sheet I will bring to the final.

3/10/00

I expect to be in my office Monday and Tuesday mornings from roughly 8:30-11:30. I may also be there at other times. If you want to be sure to catch me, send me email beforehand and ask me to confirm the time.

3/9/00

A sample change of variables problem is:
Compute the double integral of sin(2x+y) dA over the region D bounded by the lines
-x+4y=4, -x+4y=12, 2x+y=4, 2x+y=Pi.
(This was problem 5 on the review problems for the second midterm; the answer can be found here.)

3/8/00

The final is on Wednesday 3/15 in Educ 301 (our regular classroom):
  • Roughly 50% of the final will cover new material.
  • The remaining 50% of the final will consist of questions which could have been on one of the midterms.
  • The new material consists of Lessons 11-15 in the Study Guide (§13.9 & §14.6-§14.9 in the text).
    (3-d change of variables will not be tested.)
  • There is also a very nice summary of the main theorems in §14.10.
  • You should study the review sections at the end of Chapters 11, 12, 13, and 14.
  • The exam is closed book, and calculators may not be used.
  • You may bring three 3×5 index cards (both sides) OR one 8½×11 piece of paper (one side) of handwritten notes.
  • You can find a sample final (not written by me) here.
    (The solutions are here, but please try the problems without peeking first.)
  • You may also wish to review the sample problems for the previous midterms, which you can find below.
  • Most of Friday's lecture will be devoted to review.
    Come prepared to ask questions!

    3/7/00

    There is a subtlety to today's worksheet: Strictly speaking, the given vector field does not satisfy the conditions for Stokes' Theorem if the surface goes through the origin. Nevertheless, since the curl of this vector field is horizontal, it's flux upwards through the xy-plane is automatically zero -- the term whose denominator goes to zero at the origin does not appear in the final calculation! Thus, you can use Stokes' Theorem for this surface -- but you should really give a brief justification of why this case is an exception.

    3/3/00

    Tevian's first rule of vector calculus:
    When using the right-hand rule, always use your right hand...

    3/1/00

    Here are the details of the talk I am giving next week about the 1999 Nobel Prize in Physics.
    (I spent two years in the Netherlands working with one of the recipients, Professor Gerard 't Hooft.)
    This talk does not have anything to do with vector calculus.
    UNDER THE SPELL OF THE GAUGE PRINCIPLE:
    The 1999 Nobel Prize in Physics
    OSU Physics Colloquium
    4 PM, Monday, March 6, 2000, Weniger 153
    (refreshments @ 3:30 in Weniger 305)

    2/29/00

    There is a typo in the last problem on Worksheet 6, which should read x2 rather than x.

    2/28/00

  • There is a list of common surface elements in Lesson 13 in the study guide.

    2/25/00

  • There is a list of the most common types of parametric surfaces in Lesson 12 the study guide.
  • The formula for dS for the special case z=f(r,theta), given in class today, is in the study guide, but not the text.

    2/21/00

    Today's office hour is cancelled; sorry for any inconvenience.

    2/18/00

    A Java-based "microscope" for visualizing vector fields can be found here.
    This tool allows one to see the divergence and curl directly; try it!

    2/17/00

    Below are the answers to the second midterm; an answer key has been posted outside my office.
    1. 10 Pi2
    2. (a) 0 (b) 2x (c) 1
    3. (a) not conservative (b) conservative; f = x2+y+yz+xyz+C
    4. -9 Pi
    5. (a) Figure 3 (b) Figure 1 (c) Figure 2

    2/14/00

  • The midterm will not cover the two vector versions of Green's Theorem.
  • You can find some review questions (not written by me) here.
    My apologies for not including this link until now.

    2/11/00

  • Example 5 in §14.4, together with the associated problem 29, indicate the power of Green's Theorem to evaluate integrals without integration!
  • There is an important typo in the first identitiy given on the first page of Lesson 10 in the study guide, which should state that curl(grad(f)) = 0 for any function f.

    2/10/00

    If you have access to Maple, a worksheet showing how to visualize line integrals of functions of 2 variables with respect to arclength as the area of a curved "ribbon" can be found here.
    (Further information about using Maple can be found here.

    2/9/00

    The second midterm is in class on Wednesday 2/16:
  • The midterm covers Lessons 6-10 in the Study Guide (§14.1-§14.5 in the text).
  • You should also study the review section at the end of Chapter 14.
  • You can find a sample midterm (not written by me) here.
  • The exam is closed book, and calculators may not be used.
  • You may bring two 3×5 index cards (both sides) of handwritten notes;
  • Both Monday's lecture and Tuesday's recitation (Section 010 only) will be devoted to review.
    Come prepared to ask questions!

    2/4/00

    Today's lecture (Lesson 8 in the study guide) described a method for finding potential functions in 2 dimensions. Another method (for 3 dimensions, but the principle is the same) is described at the end of Lesson 10.

    2/2/00

    You may turn in your lab writeup during class on Wednesdays if it is ready by then.
    Otherwise, turn it in to Julie by 3 PM on Thursdays.

    2/1/00

    As I believe Julie pointed out in recitation, it is not necessary to use the hint in problem 1.(a) on the worksheet. Nevertheless, introducing the radial coordinate r, and noting that, for instance, the derivative of r with respect to x is just x/r, yields the indicated method for calculating the necessary partial derivatives, which you may find simpler.

    1/28/00

    Below are the answers to the first midterm; an answer key has been posted outside my office.
    1. T = 3/5 i+4/5 j ; N = 4/5 i-3/5 j ; K = 6/125
    2. 12
    3. (a) 4/5 i+3/5 j (b) 10 (c) 4/5 i+3/5 j+10 k (or any multiple)
    4. local min at (1,1)
    5. absolute min of -2 at (1,1) ; absolute max of 16 at (-2,-2)

    1/25/00

    Answers to the review problems are:
    1. r = (2t-sin(t)) i+(2t-cos(t)+1) j+(2et-t-2) k
    2. (a) 5 (b) (8 i+9 j+2 k)/1491/2 ; 1491/2 (c) 8x+9y+2z = 25
    3. (a) v = 3et i+(2t+4) j ; v = (9e2t+4(t+2)2)1/2; a = 3et i+2t j (b) v = 3 i+4 j ; v = 5 ; a = 3 i+2 j
        (c) T = 3/5 i+4/5 j ; N = 4/5 i-3/5 j ; K = 6/125 ; aT = 17/5 ; aN = 6/5
    4. saddle at (0,0); min at (1,1)
    5. min=1-301/2 at (-3/5,-2/5)*301/2 ; max=1+301/2 at (3/5,2/5)*301/2

    1/24/00

    Midterm 1 will not cover the method of Lagrange multipliers with 2 constraints.
    Due to another commitment (I am teaching another class), I will have to cut short my Wednesday office hour this week only (1/26) just before 10 AM. The remaining hour (10-11 AM) will be covered by Julie (in Kidd 330).

    1/20/00

    The answer to the question posed at the end of yesterday's class, namely:
    Find the equation of the tangent plane to the surface 3x2+4y2+5z2 = 73 at the point (2,2,3)
    is 6x+8y+15z = 73.

    1/19/00

    The formula given in class today (and in the Study Guide) for the equation of a plane whose normal vector is given is derived in §11.5. The statement made in class (and in the Study Guide) that this also works in 2 dimensions does not appear to be in the text. The precise statement is:
    The equation of the line in the plane whose normal vector is n=ai+bj takes the form ax+by=constant.
    (The constant can be determined by plugging in any point on the line.)
    Also, since Julie is out of town this week, please turn in Worksheet 2 to me (by 3 PM Thursday). There is an envelope outside my office if I am not there.

    1/18/00

    The first midterm is in class on Wednesday 1/26:
  • The midterm covers Lessons 1-5 in the Study Guide (§11.7-§11.9 and §12.6-12.8 in the text).
  • You should also study the review sections at the end of Chapters 11 and 12.
  • You can find some review questions (not written by me) here.
  • The exam is closed book, and calculators may not be used.
  • You may bring a 3×5 index card (both sides) of handwritten notes;
  • Bring your OSU ID to the exam!
  • Both Monday's lecture and Tuesday's recitation (Section 010 only) will be devoted to review.
    Come prepared to ask questions!

    1/14/00
    In each case below, the first picture shows the (3-d!) graph of a function z = f(x,y), and the second shows the combined (2-d!) graph of the level curves and gradient of f.
    (You might obtain better quality pictures using an external JPEG viewer rather than your browser.)
    paraboloid: graph gradient
    saddle: graph gradient

    1/12/00
    Here is an example of a function of 2 variables with 2 local maxima but no local minima -- something which is not possible for (continuous) functions of 1 variable.
    (The graph shown is z = - (x2-1)2 - (yx2-x-1)2.)

    1/5/00 (Section 010 only)

    Lab writeups will be due to the TA by 3 PM on the Thursday following the lab. There should be an envelope outside her office in case she is not there.