Fall 2019 - Math E-21a weekly topics and assignments

Assignments will be updated as the course proceeds. Check back each week after class.
[last updated Monday, December 23, 2019 1:31 PM]

Date Topics Homework assignments (4th Ed.) [Solutions][underlined = graded]
Thurs, Sept 5
Class #1

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.

Lecture #1 Notes

 HW#1: Read sections 9.1 to 9.3 of the text and do problems: To be turned in Sept 12: [Online deadline Sept 14] 9.1/8,12,14,16,21-27,28-32,38 9.2/22,28,38,43 Challenge Problem (see PDF) For additional practice: 9.1/7,11 9.2/15-18 HW #1 problems (PDF)
Thurs, Sept 12
Class #2

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.

Summary notes on dot products and cross products

Lecture #2 Notes

 HW2: Read sections 9.3 - 9.5 and do the following: To be turned in Sept 19: [Deadline (DL) Sept 21] 9.3/20,30,31,41,43,46 9.4/20,22,24,33 9.5/2,5,10,18,25,27,32,56,58 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: 9.3/21,24 9.4/7,11,27,36 9.5/21 HW #2 problems (PDF)
Thurs, Sept 19
Class #3

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.

Lecture #3 Notes

Supplement on parameterized curves

 HW3: Read sections 9.6, 10.1 and 10.2 and do problems: To be turned in Thurs, Sept 26: [DL Sept 28] 9.6/11,12,34 Ch 9Rev/8,10,25,26,36 (pgs. 689-690) Ch 9 T/F (pg. 688) 10.1/6,7,8,9,37,44 10.2/4,6,24,32 For additional practice: 9.6/13,15,23 10.1/19-24,25,27 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 26
Class #4

Velocity and acceleration vectors (10.2 and 10.4); arclength, unit tangent vector, curvature, unit normal vector (10.3). Functions of several variables; graph of a function of two variables (11.1); partial derivatives and differentiability (11.3).

Lecture #4 Notes

 HW4: Read the relevant portions of sections 10.2-10.4 and 11.1-11.3. (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 3: [DL Oct 5] 10.2/16,20,52;   10.3/4,6,20,22;   10.4/8,36;   11.1/5,6,20,27; 11.3/16,20,25,36,39,40,76 For additional practice: 10.2/43,49;   10.3/1,19,21 10.4/4,6,7 11.1/3, 35-40 HW #4 problems (PDF)
Thurs, Oct 3
Class #5

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); gradients and normal vectors (11.6); implicit differentiation (11.5).

Lecture #5 Notes

Practice Exam #1     Solutions

 HW5: Read the relevant portions of sections 11.1-11.6 and do: Due Thurs, Oct 10: [DL Oct 12] 11.2/11,14 11.4/6,33,37,44 11.5/2,36,38 11.6/8,10,12,16,32 For additional practice: 11.2/10,13; 11.3/10,11;   11.4/1,19,31 11.5/1,33,37 11.6/7,15,31 HW #5 problems (PDF)
Thurs, Oct 10
Class #6

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); quadratic approximation (11.7).

Midterm Exam 1     Solutions
(covering topics from Lecture #1 - Lecture #5)

 HW6: Read the relevant portions of sections 11.2-11.6 and do: Due Thurs, Oct 17: [DL Oct 19] 11.3/48,54,68 11.5/22,26,32,43,53 11.6/44,50,52 For additional practice: 11.3/45,51,61 11.5/21,51 11.6/35,41,51 HW #6 problems (PDF)
Thurs, Oct 17
Class #7
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); Method of Least Squares; constrained optimization and the Method of Lagrange Multipliers (11.8). Extreme values of a continuous function defined on a bounded region.

Lecture #7 Notes

 HW7: Read sections 11.7 and 11.8 and do problems: To be turned in Thurs, Oct 24: [DL Oct 26] 11.7/6,10,12,14,28,36,38,39,43,44,46,52 Least Squares problem (see PDF) 11.8/4,8,28,31,35 For additional practice: 11.7/3,4,5,7,27,51 [You MUST use unconstrained optimization methods for problems 11.7/36,39,43.] HW #7 problems (PDF)
Thurs, Oct 24
Class #8

Extreme values of a continuous function defined on a bounded region; Method of Lagrange Multipliers, continued; examples of unconstrained and constrained optimization in economics; multiple constraints. Introduction to integration over regions in R2.

Lecture #8 Notes

HW #8 problems (PDF) - due Thurs, Oct 31 (DL Nov 2)
(includes 11.8/16,18,23-24,26,40,42,46 and 12.1/4,12 and more)
(see solutions for which problems were graded)

Thurs, Oct 31
Class #9
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.

Lecture #9 Notes

 HW9: Read sections 12.2-12.5 and 12.7 and do problems: To be turned in Thurs, Nov 7: [DL Nov 9] 12.2/12,20 12.3/8,30,32,48,50 12.4/10,20,24,26,36 12.5/2,6,12 For additional practice: 12.2/3,9,35 12.3/4,6,7,21,24,25,28,31,47 12.4/9,18,19,21,25,29 12.5/1,5,11 HW #9 problems (PDF)
Thurs, Nov 7
Class #10
Triple integrals, continued. Applications of triple integrals: average value, centroid, mass, center of mass, weighted averages, moment of inertia. Triple integrals in cylindrical and spherical coordinates.

Lecture #10 Notes

 HW10: Read sections 12.7, 12.8, and 12.9 and do problems: To be turned in Thurs, Nov 14: [DL Nov 16] 12.7/12,16,18,22,38,46 12.8/10,12,16,26,28,31 For additional practice: 12.7/4,11,17,21,34,37,45,51 12.8/7,11,13,19,35,37 HW #10 problems (PDF)
Thurs, Nov 14
Class #11

General change of variables in multiple integrals. 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.

Lecture #11 Notes

Supplement on Conservative Vector Fields and the Fundamental Theorem of Line Integrals

Topics and Exam #2 Practice Problems     Solutions

 HW11: Read sections 12.9 (change of variables in multiple integrals), 13.1-13.4 (vector fields, line integrals, the Fundamental Theorem of Line Integrals, and Green's Theorem). To be turned in Thurs, Nov 21: [DL Nov 23] 12.9/16,17 13.1/6,25 13.2/2,7,10,20,28,34 13.3/8,10,12,16,20 (see Supplement) For additional practice: 12.9/19,21 13.1/(11-14),(15-18),(29-32) 13.2/1,14,15,17,27 13.3/19 HW #11 problems (PDF) and Supplement
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.
Thurs, Nov 21
Class #12

Fundamental Theorem of Line Integrals, independence of path, conservative vector fields, potential functions; Green's Theorem; divergence and curl of a vector field.

Lecture #12 Notes

Midterm Exam 2     Solutions

Lecture and the exam took place in Science Center Hall C.

 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, and Supplement on Curvilinear Coordinates - Div, Grad, Curl, and the Laplacian. To be turned in Thurs, Dec 5: [DL Dec 7] 13.4/4,6,10,12,18,22,24 13.5/2,14,18,20,26,38 For additional practice: 13.4/3,13 13.5/1,15,16,25,29,37 HW #12 problems (PDF) [An extra-credit Challenge Problem is included. Turn in separately.] Note on the Extra Credit Problem: No credit will be given for merely quoting a related theorem from Internet sources or illustrating it with an isolated example. An actual proof (using Green's Theorem) is necessary for any extra credit. Also, your proof must have sufficient clarity.
No class on Thurs, Nov 28 due to Thanksgiving Holiday
Thurs, Dec 5
Class #13

Parameterized surfaces in R3; integration on parameterized surfaces; surface area. Calculation of surface integrals; flux of a vector field through a surface. Geometric definitions of divergence and curl, and derivation of algebraic definitions from the geometric definitions. Statement of the Divergence Theorem and Stokes' Theorem.

Lecture #13 Notes

Supplement on surface integrals - toolkits for spheres, cylinders, graphs, and any parameterized surface.

 HW13: Review section 10.5 (parameterized surfaces) and read sections 12.6 (surface area), 13.6 (surface integrals), the Supplement on surface integrals, and the theorems in 13.7-13.8. To be turned in Thurs, Dec 12: [DL Dec 14] 12.6/2,4,6,22,26 13.6/10,16,24,26,30,38 13.8/2,4 For additional practice: 12.6/3,10,11,23,27 13.6/9,11,13,14,15,17,18,21,23,27,37,40,46 HW #13 problems (PDF)
Thurs, Dec 12
Class #14

Geometric definitions of divergence and curl, and derivation of algebraic definitions from the geometric definitions (time permitting). Sketch of proofs of the Divergence Theorem and Stokes' Theorem from these geometric definitions. Proof of Green’s Theorem from Stokes’ Theorem.

Lecture #14 Notes

Practice Final Exam     Solutions

No notes are permitted on the Final Exam. Calculators are OK, but only for ordinary calculations, i.e. you should actually carry out any necessary integrations.

 HW14: Review section 10.5 (parameterized surfaces) and read sections 12.6 (surface area), 13.6 (surface integrals), the theorems in 13.7-13.8, and the Supplement on surface integrals. Do these problems, but don't turn them in (solutions will be posted): 13.7/7,10,13,14 13.8/6,8,10,18 For additional practice: 13.7/9,15 13.8/1,3,7,9,17 HW #14 problems (PDF) - Do them, but don't turn in. Mega-List of Math E-21a techniques       Math E-21a Useful Facts
Thurs, Dec 19
8:00pm
The Final Exam will take place on Thursday, December 19 from 8:00pm to 10:00pm in Science Center Hall C. This will be a 2-hour exam. There will be no special arrangements for alternate times or locations for the Final Exam except for students with disabilities. Any medical exceptions must be requested directly via the Extension School. Distance students should make the necessary arrangements through the Exams Office.