MATH 251: Multivariable calculus. Sections 08,09,10. Spring 2008.

General Information
Lecturer: Eduardo González
Office: 342, Hill Center.
Office Phone: 732-445-2656
e   m a i l:
Office hours: Tuesday, 1:30-3:00pm.
Meeting: TuF 12:00-1:20 pm PH 111 BUS
Section 8 - W3 12:00-1:20 - ARC 204
Section 9 - W4 1:40-3:00 - SEC 208
Section 10 - W5 3:20-4:40 - SEC 206
Final Exam: TBA
Text: James Stewart, Calculus: Early Transcendentals, 5th edition Brooks/Cole, 1999, ISBN 0-534-39321-7.
General Web page:

http://www.math.rutgers.edu/courses/251/251-s08/

Special Announcements

Final exam: May 8th, 12pm-3:00pm in usual lecture hall.
Formulas for the final (will be provided): PDF.

Exams: There will be two mid-term exams and a cumulative final. The final will count 200 points. Each midterm will count 100. They will be closed book exams and student-prepared formula sheets will not be permitted.

Homework and quizzes: Homework problems will be assigned each week and collected. Homework will count 50 points toward the term grade. Quizzes will be given they will count 50 points.

Maple lab: Several maple labs will be assigned. They will be collected two weeks after they are assigned. Visit the general webpage for more information and help on Maple.

Make-up exams, late homework/Maple-lab policy: There will be no make-up exams, except in case of documented emergency. Late homework or Maple labs will not be accepted.

In summary, here are the components of the term grade with their maximum possible points:

 Component Points Hour Exams 200 Final 200 Quizzes 50 Homework 50 Maple labs 50 Total 550

Suggested homework
There is a list of suggested problems in http://www.math.rutgers.edu/courses/251/homework.html The actual homework will be assigned during class.

Homework 1. Due in recitation.
12.1; 3,17,31
12.2; 11,25,31
12.3; 1,17,27
12.4; 9,31,45

Homework 2. Due in recitation.
12.5; 3,13,17,19
13.1; 1,11,19,23
13.2; 5,11,17,31

Homework 3. Due in recitation.
13.3; 3, 11, 13, 19, 23, 27, 39
13.4; 13, 21, 25, 35
14.1; 5, 9, 27, 28, 43, 45, 53, 55, 57, 59

Homework 4. Due in recitation.
14.2; 3, 7, 13, 27, 35, 37
14.3; 7, 9, 15, 17, 41, 45, 47, 57, 67, 83
14.4; 5, 11, 19, 23, 33
14.5; 3, 7, 17, 25, 29, 39, 43, 45

Homework 5. Due in recitation.
14.6; 4, 9, 21, 31, 33
14.7; 3, 7, 13, 39

Homework 6. Due in recitation.
14.8; 3, 5, 9, 19
15.1; 8,3
15.2; 1, 9, 17, 23, 29
15.3; 1, 5, 7, 17, 21, 41, 43

Homework 7. Due in recitation.
15.4; 5, 9, 17, 21, 23, 25, 31
15.6; 1, 3, 7
15.7; 3, 9, 13, 17, 19, 25, 31, 35
15.9; 5, 7, 9, 11, 13, 21

Homework 8. Due in recitation.
15.8; 1, 3, 13, 17, 21, 29, 33, 35
16.1; 1, 5, 11, 13, 17, 21, 29, 33
16.2; 1, 5, 15, 17, 19, 21, 31, 37

Homework 9. Due in recitation.
16.3; 7, 9, 13, 15, 19, 23, 29, 31, 33
16.4; 3, 9, 11, 15, 17, 19
16.5; 3, 5, 13, 15, 25

Maple Labs
Before you do the Labs, please review the help page.

Maple Lab 4.Background information. Due 04/16/08. You are encouraged to discuss this assignment with other students and with the TA, but the work you hand in should be your own. On this page is posted individualized data for each student. For this lab, thse data will information about a region in the plane, and information about a region in space:
• The region in the plane will be defined by its boundary curves. Both curves will be of the form y = f(x). One curve will be a straight line, and one curve will be the graph of a fourth degree polynomial. The area of the region will also be given.
• The region in space will be those points which are both inside a sphere centered at the origin, and above the graph of a circular paraboloid. The volume of this region will also be given. A density function, a polynomial possibly involving x and y and z , will be given.
Use Maple to help you answer the following questions. Display the region in the plane graphically. Assume the region in the plane has constant density. Where is the center of gravity (centroid) of the region? Along the way, verify the value given for the area of the region. Display the region in space graphically. Verify the value given for the volume of the region. Using the given mass density, find the total mass of the material filling the region. Please hand in the following material:
1. All pages should be labeled with your name and section number. Also, please staple together all the pages you hand in.
2. A printout of all Maple instructions you have used. (Yes, you may and should “clean up” by removing the instructions that had errors.)
3. A clear picture of the region in the plane.
4. Computation of the area of the region (verifying the given information), and the total moments about each coordinate axis. Compute the center of gravity of the region. Label the total moments and the center of gravity.
5. A clear picture of the region in space.
6. Compute the volume of the region, including appropriate specification of the intersec- tion points of the two surfaces defining the boundary.
7. Compute the total mass in the region using the given density distribution by computing a triple integral or integrals.
Maple Lab 3.Background information. Due 03/26/08. You are encouraged to discuss this assignment with other students and with the TA, but the work you hand in should be your own. On this page is posted individualized data for each student. For this lab the data will be two functions and three constants.
• The first function, F(x, y), will be a second degree polynomial of two variables. There will also be a specific value given for x, say x = a.
• The second function, G(x, y, z), will be a second degree polynomial of two variables. There will also be a value given for x and a value given for y, let’s say x = b and y = c .
Use Maple to help you answer the following questions. What kind of curve F (x, y) = 0 is? Is it a hyperbola, a parabola, or an ellipse? For which values of y is the point (a, y) on the curve F (x, y) = 0? For each of these values of y, use Maple to compute a vector normal to F (x, y) = 0. Then use Maple to draw this vector or vectors, together with the curve F (x, y) = 0. What kind of surface is G(x,y,z) = 0? Is it a cylinder, a cone, a paraboloid (what type of paraboloid?), an ellipsoid, or a hyperboloid (what type of hyperboloid?). For which values of z is (b, c, z ) on the curve G(x, y, z ) = 0? For each of these values of z , use Maple to compute a vector normal to G(x, y, z ) = 0. Then use Maple to draw this vector or vectors, together with the surface G(x, y, z ) = 0. Please hand in the following material:
• All pages should be labeled with your name and section number. Also, please staple together all the pages you hand in.
• A printout of all Maple instructions you have used. (Yes, you may and should “clean up” by removing the instructions that had errors.)
• A clear picture of F(x,y)=0 including your identification of the curve. The identifi- cation can be done “by hand” on your printout. Show evidence for your assertion.
• Indicate clearly the coordinates of the point or points (a,y) in your computations.
• A picture of the curve F(x,y)=0 together with the normal vectors. Select the picture carefully. It should show the vectors as perpendicular to the curve.
• A clear picture of G(x,y,z)=0 including your identification of the surface. You may give several pictures and select your views carefully. The identification can be done “by hand” on your printout. Show evidence for your assertion.
• Indicate clearly the coordinates of the point or points (b,c,z) in your computations.
• A picture of the curve G(x,y,z)=0 together the normal vectors. Select the picture carefully. It should show the vectors as perpendicular to the surface. You may need to give several views of this picture and select your views carefully.

Maple Lab 2 Background information
• All pages should be labeled with your name and section number. Also, please staple together all the pages you hand in.
• A printout of all Maple instructions you have used. (Yes, you may and should “clean up” by removing the instructions that had errors.)
• Identify a graph of the curve clearly in your printout. If you need to, show several graphs of the curve which will help convince the reader that your curve does or does not have several self-intersections.
• Give several graphs of the curve which allow you to identify a “box” in which the curve sits. The box should be of the form xmin ≤ x ≤ xmax , ymin ≤ y ≤ ymax , and zmin ≤ z ≤ zmax . You can indicate the dimensions “by hand” on your printout.
• Compute the length of your curve.
• Compute the curvature of the curve when t = 2. Show that this matches the information you were given. Graph the curvature of your curve as a function of the parameter, t. Indicate “by hand” on this graph the value of the parameter and the value of the curvature for the point on the curve which has the largest curvature.
• Compute the coordinates of the point on the curve which has the largest curvature. Identify this point with the greatest curvature on a graph of your curve. You may wish to show both a constrained and an unconstrained view.

Maple Lab 1. Background information. Due 02/13/08.

You are encouraged to discuss this assignment with other students and with the TA, but the work you hand in should be your own. On this page is posted individualized data for each student. For this lab, the data will consist of coordinates for three points, p, q, and r, in the space. Then pq will denote the vector directed from p to q and pr will denote the vector directed from p to r. The vector v will be pq × pr, the cross product (vector product) of the two vectors. T will be the triangle in the space whose vertices are p, q, and r. Use Maple to compute pq, pr, and v . Use Maple to sketch these three vectors and the triangle T in one picture. This assignment is due on 02/13/08. Late submissions will not be accepted.
This is what you have to hand in:
• A printout of all Maple instructions you have used. You may and should “clean up” by removing the instructions that had errors.
• All pages should be labeled with your name and section number. Also, please staple them.
• Identify clearly in your printout the components of the vectors pq, pr, and v. These identifications can be done “by hand”.
• Hand in a printout of a picture of the three vectors and the triangle T . The picture should include labeled axes and should show the geometry of the situation well. Label the points p, q, and r in your picture. Label the vector v in your picture. Label the triangle T in your picture.

Tentative Schedule

This is just a tentative schedule, the actual topics might change.

```Lecture Sections covered

1 	12.1, 12.2
2 	12.3, 12.4
3 	12.4, 12.5
4 	13.1, 13.2, Assign Maple Lab 1
5 	13.3, 13.4 (to Kepler's laws)
6 	14.1, 14.2, 12.6
7 	14.3, 14.4
8 	14.5; Assign Maple Lab 2 (Lab 1 due)
9 	14.6
10 	14.7; Review for the Exam 1
11 	Exam 1: through 14.6
12 	14.8, Assign Maple Lab 3  (Lab 2 due)
13 	15.1, 15.2
14 	15.3
15 	15.4, 15.9 (for double integrals)
16 	15.7
17 	12.7, 15.8
18 	16.1, 16.2, Assign Maple Lab 4  (Lab 3 due)
19 	16.2
20 	16.3
21 	16.4, Review for the Exam 2
22 	Exam 2: through 16.3
23 	16.5 ; Review of the exam
24 	15.6, 16.6  Maple Lab 4  due
25 	16.7
26 	16.8
27 	16.9
28 	Review for the Final Exam

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