LSU College of Science
LSU  | Mathematics

Math 2057---Sec. 3 Information

For Spring 2012

Time and Room MWF 2:40 PM--3:30 PM, Room 243 Lockett
Calendar Our class meets from Wednesday, January 18 through Friday, May 4. The Final Exam will be Friday, May 11, from 12:30 -‐ 2:30 PM
Office Hours T, Th noon -- 2:00 PM. I am available at many other times. Call first to make sure I'm able meet with you. I answer email many times daily.
Leonard Richardson Office 386 Lockett
Text Jon Rogawski, (Multivariable) Calculus---Early Transcendentals, first edition, W. H. Freeman and Co., 2008. We will cover most of Chapters 14, 15, 16, and 17. These chapters are available as a separately bound volume called Multivariable Calculus, or as part of a single larger volume without the word Multivariable in the title.
Prerequisites Mathematics 1552 or 1553.
Free Math Tutoring The Mathematics Department offers free tutoring MTWTh 10 AM -- 7 PM, and F 10 AM -- 3 PM, in Room 141 Middleton Library. Also there is Genesis Tutoring, a free tutoring service offered by the Office of Multicultural Affairs to all LSU students. For spring 2012, they say they have tutors who can assist in Math 1021; 1022; 1023; 1431; 1550; 1552; 2057. These tutors are available Monday – Thursday, 5 pm – 9 pm. Genesis Tutoring is located on the 3rd floor of the Student Union in room 335.

General Advice

Please understand that it is from the effort of working your way through assigned problems on paper that you learn mathematics. It is by no means sufficient to read solutions in a solutions manual! And although we hope you benefit from seeing how solutions are presented in class, you must not expect to learn how to solve problems just from watching. You must work out problems yourself, the hard way, in order to learn this work. Examination problems will be very similar to assigned homework problems. Thus your daily effort on homework problems will be strongly reflected in your test grades.


No books or notes are permitted. The problems will be similar to those in the homework. If you miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable! You can see questions and solutions from last year's hour tests from this course. Look below in the table showing the assignments. Caution! Do not assume that the selection of topics covered on a test will be identical to last year's selection! Even three hour tests and a two-hour final exam are insufficient to test all the topics you should learn. Thus there will be at least some variation of topics chosen from year to year.


There will be three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. I will grade your hour tests and return them to you the very next class meeting if possible. Your final test average will be sum of all your test grades divided by 5. So your final test average will be less than or equal to 100. Your final letter grade will be as follows: 90 -100 (A), 80-89 (B), 70-79 (C), 60-69 (D), Below 60 (F). You should save all your graded work for future study and in case you think your final grade is in error. For the spring 2012 semester only, your lowest hour test grade will be replaced by your final exam grade converted to a 100-ppoint scale, provided that this improves your semester average. Be sure to read the link below: Click here for a Five-Step Plan to improve your grades!

Unhappy with your grades in Math?

Click here for a Five-Step Plan to improve your grades!


It is especially important not to fall behind. It is very important to attend class and participate in homework discussions daily. Please do not assume you can take care of difficulties later---see me or the graduate assistant for help as soon as possible if there is something you do not understand! You are responsible for all assigned problems---not just those which we go over the next day!

Computer Support

The full power of Mathematica is available on many LSU computers, including those in the Math Department's computer labs and in the Library as well. But there is a simplified Web Mathematica which is free to use online if you click on the link in this sentence. If you have not already had the Math Department's course in Mathematica, you might find it simpler to figure out how to use the Web Mathematica. It is not as versatile, but the Plot3D function is there and works fast for surface graphs. There is also a function-integrator which can be fun to use without having to learn very much about Mathematica. But do remember, this is an auxiliary resource. The time you spend working on problems on paper is the most important part of homework when it comes to learning the subject. With that understanding, Mathematica can be fun and helpful too.

Homework Assignments and Downloads

The assignments in the table below will be revised daily for the current semester. When you see a valid due-date supplied in the left-hand column, then you will know the assignment is current for this semester. A star (*) next to a problem indicates that it is more challenging than the others. Be sure to reload this page from the website each time you visit, since it is updated daily!
January 18 Read this syllabus so you can ask questions about it in class. Bookmark this page and be sure to get the latest daily version each time you access it to find assignments. Obtain a copy of the correct text book so you will be ready for work.
January 20 Download Level Surfaces Example, Example of Domain and Range and Level Curve Example. Do 14.1/7--29(odd). There is an error in 14.1/27: The answer in the back of the book pertains to a slightly different function, z=|x|+|y|, from the one in the exercise, which is z=x+|y|.
January 23 Download Limit Example; Do 14.2/1, 7, 11, 18, 23, 26, 27, 29, 31--33, 35, 38. (Don't be discouraged. These can be challenging!) Please download this picture of the graph of z=x+|y|.
January 25 14.3/1, 3, 5, 7, 13, 17, 21, 25, 29, 37, 47, 53, 57, 72.
January 27 Download Nondifferentiable, noncontinuous function having partial derivatives everywhere; 14.4/3, 7, 9, 15, 17, 19, 27, 35, 39.
January 30 14.6/3, 7, 9*, 13, 15, 25, 27; and 14.5/9, 13, 17. The two Chain Rule theorems in these sections could be strengthened by including the assertion that the composition of two differentiable functions is differentiable. Download Proof of the Chain Rule, including proof of differentiability.
February 1 Download A nondifferentiable, continuous function having partial derivatives everywhere. Do 14.5/21, 25, 27, 29, 31; 14.6/50.
February 3 Download Saddle Point Example; Do 14.5/38 --- 45.
February 6 14.7/5, 7, 9, 11, 12, 15, 17, 19; 21, 25, 27, 29, 31, 35, 37, 39, 42, 43. Download Second Derivative Test Example. Download Minimum with No Boundary Example and Implicit Domain and Range Example. These are illustrations for some unassigned problems of interest.
February 8 Bring questions to review for hour test #1. Caution! Do not assume that the selection of topics covered on a test will be identical to last year's selection! Even three hour tests and a two-hour final exam are insufficient to test all the topics you should learn. Thus there will be at least some variation of topics chosen from year to year.
February 10 First Hour Test: Sections 14.1--14.7.
February 12 Download 2012 Solutions and Statistical Results for Test #1. You will find the solutions and class statistics at the end of the pdf document. Work out on paper the solutions to test problems you think you missed, following the sketches online at the link listed above. Ask questions in class Monday.
February 15 14.8/1, 3, 5, 7, 9, 15, 17*, 19, 21.
February 17 14.8/11, 13, 23, 25, 27*, 29, 31, 35. The answer section is wrong for #29: f(.5,.5,0)=.25, which is greater than 3/16, claimed by the answer section to be the maximum. This problem is interesting because the maximum occurs on the boundary curve (a triangle in space) of the surface of constraint. Thus Lagrange Multipliers fail here to yield the maximum! It is important always to check the boundary separately. Download Lagrange Multiplier Examples for One Surface and Two Intersecting Surfaces. These are illustrations for some other interesting exercises.
February 22 Download and read a careful solution to Exercise 14.8 / 29.
February 24 15.1/9, 15, 17, 19, 21, 25, 27, 31, 35, 37, 39.
February 27 15.2/3, 5, 9, 11, 17, 21, 23, 31, 33, 37, 39, 41.
February 29 15.3/1, 5, 9, 11, 13, 15, 17, 21, 23. Exercise 11 can be done with a sketch of the appropriate domain in the xy-plane underlying the solid. But it is also a good exercise to try to draw the solid. After trying seriously, download a picture for Exercise 11. Download A Volume Example for several intersecting surfaces.
March 2 15.4/1, 3, 5, 7, 9, 13, 15, 17, 21, 23, 25, 31, 35, 39, 41.
March 5 15.4/ 43, 47, 49, 51, 53, 55, 57, 59, 67.
March 7 p. 939 / 27, 31, 33, 37, 39.
March 9 15.5/1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
March 12 15.5/21, 23, 29, 31, 33, 35, 37, 38, 43, 44. In #38: The square with vertices (1,0), (-1,0), (0,1), (0,-1) is being named R in the xy-plane, whereas in the other exercises the domain in the xy-plane was called D.
March 14 Bring Questions for Review, covering Sections 14.8 -- 15.5.
March 16 Second Hour Test
March 18 Solutions and Statistical Results for Test #2, Spring 2012.
March 21 16.1/ 23, 25, 26, 29. Download Tangent Field Example and Gradient Field Example.
March 23 16.2/ 1, 5, 9, 15, 21, 23, 25, 31, 33.
March 26 16.3/1, 5, 9, 11--17, 19.
March 28 16.4/7, 9, 13, 17, 19, 21, 25, 29, 33, 37, 41, 43*. Hint: #41 is interesting. You will find the area of the part of a cone that lies above an ellipse in the xy-plane. Download Area of a Cone Cut by a Cylinder.
March 30 Download Möbius Band; 16.5/ 5, 7, 9, 13, 17, 20, 23, 25.
April 2 Chapter 16 Review Exercises / 2, 7, 9, 11, 15, 16, 20, 22, 41, 45.
April 4 17.1/1, 3, 5, 7, 9, 11, 23.
April 16 17.1/ 13, 15 17*, 19, 27, 28*. Download Arch of a Cycloid.
April 18 17.2/1, 3, 5, 7, 9.
April 20 17.2/ 11, 13*, 19, 21, 27. Typo: In problem #19, the area of the triangle in Fig. 18 is 8 times the square root of 3, not 16 as stated. The solution to the problem as a whole, in the answer section, is correct, however.
April 23 17.3/1, 3, 5, 7, 9, 11, 13, 15, 17*, 19, 21. Problem 21 Typo: The author probably meant to write a parametrization for (x, y, z) lying on the ellipsoid in terms of phi and theta, but enclosed these three parametrizing functions in an undefined function Phi instead. Problem 17 Hint: The volume integral can be evaluated with effort using rectangular coordinates and only one domain of integration. Cylindrical coordinates will be easier although two domains of integration are needed. Use symmetry about the x-axis to reduce the domain of integration by half. Observe the significance of Cos-1(-1/3) in setting the limits on the two domains, and write x=-1 as r=-Sec (theta). The answer is 18Cos-1(-1/3) + 76(2)1/2/3 in closed form. The answer section gives a numerical approximation that is off by less than .002, but the closed form of the answer is better because it is exact and because its form reveals aspects of the calculation upon which it rests.
April 25 Bring questions to review for third hour test. Be sure you understand the following main topics:
  • evaluation of line integrals by parameterization
  • evaluation of line integrals of conservative vector fields via the Fundamental Theorem
  • evaluation of surface integrals by parameterization
  • Green's theorem
  • Stokes' theorem
  • the Divergence theorem
April 27
Third Hour Test.
April 29 2012 Solutions and Statistical Results for Test #3
April 30 Review for Final Exam. It is important to review ALL the homework problems assigned in this course. For Monday, please bring review questions from Chapter 14 homework assignments.
May 2 Review for Final Exam. It is important to review ALL the homework problems assigned in this course. For Wednesday, please bring review questions from Chapter 15 homework assignments.
May 4 Review for Final Exam. It is important to review ALL the homework problems assigned in this course. For Friday, please bring review questions from Chapters 16 and 17 homework assignments. Also please bring a #2 pencil to fill out end-of-term College of Science course evaluation forms. In class today I gave an example of a differentiable function of two variables having no global extrema although there are infinitely many points that are local extrema. You can download the picture here!
May 7 -- 10 Final Exam Week Office Hours, MTWTh Noon -- 3 PM Room 386 Lockett Hall. For other times, call or email first.
Study for the Final Exam! This 200-point exam will cover the whole course in a uniform manner, so remember to review from the beginning of the course. Your final grade for the course will be the larger of the following two:
1. The grade guaranteed by the formula provided higher on this page.
2. One letter below the final exam grade.
Thus the final exam provides a safety net that supplements the calculations specified above.
Friday, May 11 Final Exam, 12:30 -‐ 2:30 PM
May 14 Download 2012 Solutions and Statistical Results for Final Exam and Final Grades