Fall 2009 - Math E-21a weekly topics and assignments
Assignments will be updated as the course proceeds. Check back each week after class.
[last updated December 13, 2009]

Date Topics Homework assignments [Solutions]
Thurs, Sept 3 Introduction to R2 and R3; points vs. vectors; sum of vectors, scalar multiplication; difference vector; length of a vector; distance between points; unit vectors. Equations for circles and spheres; vector and parametric equations of a line in R2 and in R3.
HW#1:  Read sections 9.1 to 9.3 of the text and do problems:
To be turned in Sept 10:
9.1/8,12,14,34
9.2/20,24,32,37
Extra Challenge Problem (see PDF)		 For additional practice: 
9.1/7,11,19-28
9.2/15-18
HW #1 problems (PDF)
Thurs, Sept 10

Dot product in R2, R3 and its algebraic and geometric properties. Scalar and vector projections. Equations of lines and planes in R3 and their intersections. The cross product in R3 and its algebraic and geometric properties, triple scalar product. Applications to distance, area, volume.
HW2: Read sections 9.3 - 9.5 and do the following: 

To be turned in Sept 17:

9.3/16,24,25,37,40

9.4/14,16,18,27

9.5/10,18,30,48,50 and

(a) Prove the Pythagorean Theorem using only formulas for areas of squares and right triangles.

(b) Prove the Law of Cosines using the Pythagorean Theorem.

For additional practice:

9.3/17,20,35

9.4/7,11,21,30

9.5/3,5,21,25,27

HW #2 problems (PDF)
Thurs, Sept 17

Cross product, triple scalar product, continued. Equations vs. parameterizations. Brief survey of functions, graphs, and surfaces. Vector-valued functions - parameterized curves in R2 and R3. Velocity vectors.
HW3: Read sections 9.6, 10.1 and 10.2 and do problems:
To be turned in Thurs, Sept 24: 
9.6/11,12,34
Ch 9Rev/8,10,26,36 (pgs691-2)
10.1/6,7,8,38
10.2/4,6,20,28		 For additional practice: 
9.6/13,15,23
Ch 9 T/F (pg. 691)
10.1/17-22,23,25,33
10.2/5,7

HW #3 problems (PDF)
Note: If you have access to mathematical software such as Mathematica or Matlab or Maple, you might try graphing some of the functions in section 9.6 and some of the curves in 10.1-10.2 on the computer.
Thurs, Sept 24

Velocity and acceleration vectors (10.2); arclength (10.3). Examples of parameterized surfaces in R3 (10.5) and “velocity vectors” tangent to curves on these surfaces. Functions of several variables; graphs and level curves (contours) of a function of two variables (11.1); partial derivatives (part of 11.3); tangent plane to the graph of a function (looking ahead - part of 11.4).
HW4: Read the relevant portions of sections 10.2-10.5 and all of 11.1 and 11.3. (If you have an interest in physics, you may want to check out some of the details in section 10.4 that we did not cover in class.) 
To be turned in Thurs, Oct 1: 
10.2/38,46;  10.3/2,4;
10.4/8,12,34;  10.5/17,22,24;
Ch 10Rev/3 (pg. 735)
11.1/6,16,23;  11.3/16,20,26,35		 For additional practice: 
10.2/39,43,44;  10.3/1;
10.4/4,6,7;  10.5/1,2,11-16;
11.1/3,15,21,31-36;  11.3/8,9 

HW #4 problems (PDF)
Thurs, Oct 1

Limits and continuity (11.2). Partial derivatives and differentiability (11.3); linear approximation; tangent plane to the graph of a function of two variables (11.4). Rate of change of a function along a parameterized curve, the basic Chain Rule, directional derivatives and the gradient vector (11.5 and 11.6); using the gradient to find normal vectors to curves and surfaces (11.6).

Exam #1 topics and Practice Exam #1      Solutions
HW5: Read the relevant portions of sections 11.2-11.6 and do:
To be turned in Thurs, Oct 8: 
11.2/10,11
11.3/37,68
11.4/4,28,31,38
11.5/2,32,34
11.6/8,10,12,14,30		 For additional practice:
11.2/13
11.4/1,13,15,25
11.5/1,29,33
11.6/7,13,29

HW #5 problems (PDF)
Thurs, Oct 8

Gradients and normal vectors (11.6); General Chain Rule (11.5); implicit differentiation (11.5); higher order partial derivatives (11.3); equality of mixed partial derivatives - Clairaut's Theorem (11.3).
Midterm Exam 1    Solutions
HW6: Read the relevant portions of sections 11.2-11.6 and do:
To be turned in Thurs, Oct 15: 
11.3/42,48,62
11.5/18,22,28,37
11.6/38,44,46		 For additional practice:
11.3/41,47,57
11.5/17,45,47
11.6/33,35,45

HW #6 problems (PDF)
Thurs, Oct 15 Quadratic approximation (11.7); unconstrained optimization - finding maximum and minimum values of functions of two or more variables, 2nd Derivative Test for stationary points of a function of two variables (11.7); constrained optimization and the Method of Lagrange Multipliers (11.8). Extreme values of a continuous function defined on a bounded region.
HW7: Read sections 11.7 and 11.8 and do problems:
To be turned in Thurs, Oct 22: 
11.7/6,10,12,14,26,34,36,37,39,42,44,50
11.8/4,8,26,31		 For additional practice: 
11.7/3,4,5,7,25,41,49

[Use unconstrained optimization methods for problem 11.7/34 and 11.7/39.]
HW #7 problems (PDF)
Thurs, Oct 22 Method of Lagrange Multipliers, continued; examples of unconstrained and constrained optimization in economics; Method of Least Squares; multiple constraints. Introduction to integration over regions in R2.

HW8: Read sections 11.8 and 12.1 (and maybe read ahead in 12.2-12.3).

HW #8 problems (PDF) - due Thurs, Oct 29
(includes 11.8/16,18,21-22,24,38,40,44 and 12.1/4,12 and more.
Thurs, Oct 29 Multiple integrals in the calculation of area, volume, mass, population, and average value of a function. Iterated integrals; changing order of integration in an iterated integral; and the Fubini Theorem. Use of polar coordinates in calculating double integrals. Geometric center of a region (centroid) and center of mass. Triple integrals in Cartesian coordinates and calculation by successive slicing.
HW9: Read sections 12.2-12.5 and 12.7 and do problems: 
To be turned in Thurs, Nov 5: 
12.2/12,18
12.3/8,22,24,26,40,42
12.4/12,18,20,22,24,32
12.5/2,6,12		 For additional practice: 
12.2/3,9,31
12.3/4,6,7,15,18,19,25,39
12.4/11,17,19,23,27
12.5/1,5,11
HW #9 problems (PDF)
Thurs, Nov 5 Triple integrals, continued. Applications of triple integrals: average value, centroid, mass, center of mass, and weighted averages. Triple integrals in cylindrical and spherical coordinates. Change of variables in multiple integrals.
HW10: Read sections 12.7, 12.8, and 12.9 and do problems:
To be turned in Thurs, Nov 12: 
12.7/4,10,12,16,20,36,40
12.8/10,12,16,22,24,27		 For additional practice: 
12.7/9,15,19,32,35,39,47
12.8/7,11,13,19,31,33

HW #10 problems (PDF)
Thurs, Nov 12

Vector fields in R2 and R3; line integrals and work done by a variable force along a parameterized curve. Calculation of line integrals; Fundamental Theorem of Line Integrals, independence of path, conservative vector fields. Test to determine if a vector field is conservative.

Exam #2 topics and Practice Exam #2      Solutions

There's a good java-based tool for showing vector fields and flows in R2 at http://math.rice.edu/~dfield/dfpp.html. Choose the PPLANE option. You can enter new x and y component functions for the vector field or change the size of the window. To see a trajectory (flow), just click on a point in the phase-plane.
HW11: Read 13.1-13.4 (vector fields, line integrals, the Fundamental Theorem of Line Integrals, and Green's Theorem).

To be turned in Thurs, Nov 19:
12.9/12,13
13.1/6,25
13.2/2,5,8,18,22
13.3/4,10,12,16,20

For additional practice:
12.9/15,17
13.1/(11-14),(15-18),(29-32)
13.2/1,12,13,15,21,28
13.3/19

HW #11 problems (PDF)
Thurs, Nov 19 Green's Theorem; equivalent statements about conservative vector fields; divergence and curl of a vector field. Test to determine if a vector field is conservative and how to find potential functions.

Midterm Exam 2     Solutions
HW12: Read sections 13.4-13.5 on Green's Theorem and curl and divergence of a vector field and test for a vector field to be conservative.

To be turned in Thurs, Dec 3:
13.4/4,8,12,14,18,22,24
13.5/2,12,16,18,24,36

For additional practice:
13.4/3,15
13.5/1,13,14,23,27,35

HW #12 problems (PDF) - This assignment was modified from the original.
[An extra-credit Challenge Problem is included.]
Thurs, Dec 3

Parameterized surfaces in R3; integration on parameterized surfaces; surface area. Calculation of surface integrals by parameterization and by using available coordinates for spheres, cylinders, projectable surfaces and graphs, and general method for any parameterized surface. Flux of a vector field through a surface. Statement of five versions of the Fundamental Theorem of Calculus, including Stokes' Theorem and the Divergence Theorem.

Supplement on integration on surfaces - toolkits for spheres, cylinders, graphs, and any parameterized surface (this may be expanded later and better drawings included).
HW13: Review section 10.5 (parameterized surfaces) and read sections 12.6 (surface area), 13.6 (surface integrals), 13.7, and the surface integration supplement.
To be turned in Thurs, Dec 10:
12.6/2,4,6,10,24,28
13.6/8,12,14,16,22,24,26,34		 For additional practice: 
12.6/3,9,23,25
13.6/7,9,11,13,15,19,21,25,33,36,42

HW #13 problems (PDF)
Thurs, Dec 10 Examples. Geometric definitions of divergence and curl, and derivation of algebraic definitions from the geometric definitions. Proofs of the Divergence Theorem and Stokes' Theorem from these geometric definitions. Proof of Green’s Theorem from Stokes’ Theorem.

Practice Final Exam problems     Solutions
HW14: Read sections 13.5-13.9.
Do these problems, but don't turn them in (solutions will be posted):
13.7/10,14
13.8/2,4,6,8,10,18		 For additional practice:
13.7/7,9,13,15
13.8/1,3,7,9,17

HW #14 problems (PDF)
Mega-List of Math E-21a techniques       Math E-21a Useful Facts
Thurs, Dec 17 FINAL EXAM in Emerson Hall Room 105  

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