Fall 2024 - Math E-21a weekly topics and assignments

Assignments will be updated as the course proceeds. Check back each week after class.
[last updated Sunday, December 22, 2024 11:58 AM]

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

Introduction to R2 and R3; points vs. vectors; sum of vectors, scalar multiplication; coordinate-free vector proofs; 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 no later than Sat, Sept 14:
9.1/8,12,14,16,21-27,28-32,38
9.2/22,28,38,43
+ two additional “vector proofs” (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

(You may also find the Lecture #3 Notes notes helpful.)
HW2: Read sections 9.3 - 9.5 and do the following:

To be turned in no later than Sat, Sept 21:

1. Prove the Pythagorean Theorem using only formulas for areas of squares and right triangles.

2. Prove the Law of Cosines using the Pythagorean Theorem.

3. Prove using vector methods (without coordinates) that an angle inscribed in a semicircle is a right angle.

and the following problems from the Stewart text::
9.3/20,30,41,43,46
9.4/20,22,24,33
9.5/10,18,25,27,32,56,58

 For additional practice:
9.3/21,24,31
9.4/7,11,27,36
9.5/2,5,21
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, calculus of vector-valued functions.

Lecture #3 Notes

Summary notes on dot products and cross products

Supplement on parameterized curves
HW3: Read sections 9.6, 10.1 and 10.2 and do problems:
To be turned in no later than Sat, 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
10.2/4,24,32		 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. Alternatively, try Wolfram Alpha: [2D curve plotter] [3D curve plotter] [Level Curve Grapher] [Level Surface Grapher]
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 no later than Sat, 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).

Lecture #5 Notes

Exam #1 (covering topics from Lecture #1 - Lecture #5) will take place in Canvas/Proctorio during a 25 hour window opening after our regular class at 11:00pm on Oct 10 and closing at 11:59pm on Fri, Oct 11. The exam will take approximately 70-75 minutes with additional time for downloading, scanning, uploading, etc. for a total allotted time of 90 minutes. The Proctorio Setup Quiz in Canvas should be done prior to taking the exam to make sure that your Chrome browser is updated and that the Proctorio extension is properly installed. Calculators will be permitted, but no other notes, texts or other aids.

Practice Exam #1     Solutions
[use same username/password as HW solutions]
HW5: Read the relevant portions of sections 11.1-11.6 and do:
To be turned in no later than Sat, 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); partial derivatives in the case of non-independent variables (w/constraints); higher order partial derivatives (11.3); equality of mixed partial derivatives - Clairaut’s Theorem (11.3).

Lecture #6 Notes

Midterm Exam 1 online in Proctorio    Exam #1 solutions
(covering topics from Lecture #1 - Lecture #5)
HW6: Read the relevant portions of sections 11.2-11.6 and do:
To be turned in no later than Sat, Oct 19:
11.3/48,54,68
11.5/22,26,32,43,51
11.6/44,50,52
+ Non-independent variables problem		 For additional practice:
11.3/45,51,61
11.5/21,53
11.6/35,41
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 no later than Sat, Oct 26:
11.7/6,10,14,28,36,38,39,43,44,46,52
Least Squares problem (see PDF)
11.8/4,8,28,31,35		 For additional practice:
11.7/3,4,5,7,12,27,51
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

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

To be turned in no later than Sat, Nov 2:
(includes 11.8/16,18,23-24,26,40,42,46 and 12.1/4,12 and more)

HW #8 problems (PDF)
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.

Lecture #9 Notes

Geographic Center of the Contiguous United States: Lebanon, Kansas
(Atlas Obscura)
HW9: Read sections 12.2-12.5 and 12.7 and do problems:
To be turned in no later than Sat, 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
12.3/4,6,7,21,24,25,28,47
12.4/9,18,19,21,25
12.5/1,5
HW #9 problems (PDF)
Thurs, Nov 7
Class #10		 Triple integrals in Cartesian coordinates and calculation by successive slicing. Applications of triple integrals: average value, centroid, mass, center of mass, weighted averages, moment of inertia. Triple integrals in cylindrical and spherical coordinates. General change of variables in multiple integrals.

Lecture #10 Notes
HW10: Read sections 12.7, 12.8, and 12.9 and do problems:
To be turned in no later than Sat, 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, Jacobian determinants. Vector fields in R2 and R3. Integration along a curve. Line integrals and work done by a variable force along a parameterized curve. Calculation of line integrals.

Lecture #11 Notes

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

Exam #2 Topics and Practice Exam     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) and Supplement.
To be turned in no later than Sat, Nov 23:
12.9/16,17,20,22,25
13.1/6,25
13.2/2,7,10,15,20,28,34		 For additional practice:
12.9/19,21
13.1/(11-14),(15-18),(29-32)
13.2/1,14,17,27
HW #11 problems (PDF)
Thurs, Nov 21
Class #12

Fundamental Theorem of Line Integrals, independence of path, conservative vector fields, test to determine if a vector field is conservative, potential functions; Green’s Theorem; divergence and curl of a vector field.

Lecture #12 Notes

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

Exam #2    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, and Supplement on Curvilinear Coordinates - Div, Grad, Curl, and the Laplacian.
To be turned in no later than Sat, Dec 7:
13.3/8,10,12,16,20 (see Supplement)
13.4/4,6,10,12,18,22,24
13.5/2,14,18,38		 For additional practice:
13.3/19
13.4/3,13
13.5/1,15,16,20,26,25,29,37
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.
  Here's a website that has a good java-based tool for showing vector fields and flows: https://www.cs.unm.edu/~joel/dfield/ The current version requires you to download a Java executable file to your own computer and to run it locally on your own machine. You can customize various options. You can also print the graphs. [How Java is called varies on computer and operating system, so we may have to provide some additional documentation.] [Also available here]
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 no later than Sat, Dec 14:
12.6/2,6,22,26
13.6/10,16,24,26,30,38, two extras
13.8/2,4, one extra		 For additional practice:
12.6/3,10,11,23,27
13.6/11,14,15,17,23,27,37,40,46
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
(try all the problems first before consulting the solutions!)

Some “Useful Facts” will be provided on the Final Exam, e.g. definitions of curl and divergence, statements Green’s Theorem, Divergence Theorem, and Stokes’ Theorem. No additional 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 in preparation for the Final Exam,
but don’t turn them in. Solutions are now posted.
Mega-List of Math E-21a techniques     Math E-21a Useful Facts
Thurs, Dec 19 A two-hour Final Exam will take place in Proctorio during a 24-hour window on Dec 19. Some additional time will be allotted for download, scanning, and uploading the complete exam. Any medical exceptions must be requested directly via the Extension School. [There will be no regular lecture during the week of the Final Exam.] The exam window will open at 12:00am on Thurs, Dec 19 and close at 11:59pm on Thurs, Dec 19.
Note: Per Harvard Extension School rules, only students taking the course for credit may take the Final Exam.

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