Math 21D: Calculus (Fall, 2019)

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Announcements

Solutions to the final are here.

The final exam will be Monday, December 9, 3:30–5:30 p.m. Depending on the first letter of your last name you should go to the following classrooms.

• A-S: Giedt 1001
• T-Z: Roessler 55

I will hold a review session: Friday, December 6, 5:10-7 p.m., 2205 Haring.

The Galois Group (the math graduate student organization) also has a tutoring fundraiser this weekend.

The final will cover all the material in the class.

• 15.1: Double integrals on rectangles. Definition as limits of Riemann sums. Evaluation as iterated integrals.
• 15.2: Double integrals on general regions. Definition as limits of Riemann sums. Interpretations of double integrals as areas, volumes, mass, charge, population, probability etc. Properties of the integral (linearity, monotonicity and additivity). Evaluation as iterated integrals. Determination of the limits of integration. Exchange in the order of integration.
• 15.3: Areas and average values by double integration.
• 15.4: Double integration in polar form. Polar coordinates. Definition of integrals as limits of Riemann sums in polar coordinates. Area element dA = rdrdθ. Determination of the limits of integration. Transformation of double integrals from Cartesian to polar coordinates. Areas in polar coordinates.
• 15.5: Triple integrals in Cartesian coordinates. Definition of integrals as limits of Riemann sums. Interpretations and properties of the integral. Evaluation as iterated integrals. Determination of the limits of integration. Exchange in the order of integration for iterated triple integrals.
• 15.7: Triple integrals in cylindrical and spherical coordinates. Cylindrical and spherical coordinates. Definition of integrals as limits of Riemann sums. Volume elements dV = rdrdθdz (cylindrical) and dV = ρ²sin φdρdφdθ (spherical). Transformation of triple integrals from Cartesian to cylindrical and spherical coordinates. Applications to volume, average value, mass etc.
• 15.8: Change of variables in multiple integrals and Jacobians.
• 13.3: Parametric equations of curves. Velocity and unit tangent vectors. Arclength of curves. Parametrization by arclength.
• 13.4: Curvature and normal vector of a curve.
• 13.5: Torsion and binormal vector of a curve. The TNB frame. Tangential and normal components of acceleration.
• 16.1: Line integrals of scalar fields with respect to arclength along curves.
• 16.2: Line integrals of vector fields. Arclength, parametric, vector, component, and differential form expressions for line integrals. Work done by forces. Circulation of a vector field around a curve. Flux of a vector field across a curve in the plane.
• 16.3: Conservative vector fields. Equivalence of gradient vector fields and path-independent vector fields. Line integral of a gradient field is the difference of the potential between the endpoints. Zero curl condition for a gradient vector field and determination of the potential from the vector field. Equivalence of gradient vector fields and exact differential forms.
• 16.4: Positively oriented boundary of a region in the plane. Green's theorem in the plane. Circulation-curl and flux-divergence interpretations.
• 16.5: Surfaces and areas.
• 16.6: Surface integrals.
• 16.7: Stokes's theorem.
• 16.8: Divergence theorem.

For practice, here are the midterm exams from the other 21D class this quarter, courtesy of Prof. Freund:

• Midterm 1 (you can ignore Problem 6)
• Midterm 2 (you can ignore Problem 5)
• Midterm 3

There won't be any sample problems for the material on surface integrals and the Stokes and divergence theorems. You should work over the homework problems for those sections.

Solutions to these midterms can be found on Prof. Freund's 21D webpage here.

Important Dates

• Instruction begins: Wednesday, Sept 25
• Last day to add: Thursday, October 10
• Last day to drop: Tuesday, October 22
• Last class: Friday, December 6
• Academic holidays: Monday, November 11; Thursday, November 28; Friday November 28

TA information

Lead TA: Brian Harvie
e-mail: bharvie@math.ucdavis.edu

Discussion sections:

• B01 HOAGLD 108 R 0610-0700 PM, Christopher Alexander
• B02 HOAGLD 108 R 0510-0600 PM, Christopher Alexander
• B03 YOUNG 184 R 0810-0900 PM, Norman Shue
• B04 WELLMN 230 R 0810-0900 PM, Yuan Ni
• B05 HOAGLD 108 R 0710-0800 PM, Norman Shue
• B06 CHEM 166 R 0810-0900 PM, Brian Harvie
• B07 OLSON 101 R 0510-0600 PM, Yuan Ni

TA Help: The Calculus Room in MSB 1118 is open 10a.m.–7p.m. Mon. to Thu., and 10a.m.–6p.m on Fri.

Exams

There will be two in-class midterms and a final. There will be no makeup exams.

• Midterm 1: Friday, October 18
• Midterm 2: Friday, November 22
• Final: Monday, December 9, 3:30–5:30 p.m. (location TBA)

No notes, books, or electronic devices are allowed in any exams.

Midterm 1

Solutions to Midterm 1 are here.

The first midterm will be in class on Friday, Oct 18. All exams are closed book (no notes or electronic devices). The midterm will cover multivariable integration.

• 15.1: Double integrals on rectangles. Definition as limits of Riemann sums. Evaluation as iterated integrals.
• 15.2: Double integrals on general regions. Definition as limits of Riemann sums. Interpretations of double integrals as areas, volumes, mass, charge, population, probability etc. Properties of the integral (linearity, monotonicity and additivity). Evaluation as iterated integrals. Determination of the limits of integration. Exchange in the order of integration.
• 15.3: Areas and average values by double integration.
• 15.4: Double integration in polar form. Polar coordinates. Definition of integrals as limits of Riemann sums in polar coordinates. Area element dA = rdrdθ. Determination of the limits of integration. Transformation of double integrals from Cartesian to polar coordinates. Areas in polar coordinates.
• 15.5: Triple integrals in Cartesian coordinates. Definition of integrals as limits of Riemann sums. Interpretations and properties of the integral. Evaluation as iterated integrals. Determination of the limits of integration. Exchange in the order of integration for iterated triple integrals.
• 15.7: Triple integrals in cylindrical and spherical coordinates. Cylindrical and spherical coordinates. Definition of integrals as limits of Riemann sums. Volume elements dV = rdrdθdz (cylindrical) and dV = ρ²sin φdρdφdθ (spherical). Transformation of triple integrals from Cartesian to cylindrical and spherical coordinates. Applications to volume, average value, mass etc.

Section 15.6 (on centers of mass and moments of inertia) won't be covered by the exams. Section 15.8 (on change of variables) will be covered in subsequent exams.

Sample midterm problems are here.

Solutions to the sample midterm problems are here.

Midterm 2

Solutions to Midterm 2 are posted here.

The second midterm will be in class on Friday, Nov 22. All exams are closed book (no notes or electronic devices). The midterm will cover vector calculus.

• 13.3: Parametric equations of curves. Velocity and unit tangent vectors. Arclength of curves. Parametrization by arclength.
• 13.4: Curvature and normal vector of a curve.
• 13.5: Torsion and binormal vector of a curve. The TNB frame. Tangential and normal components of acceleration.
• 16.1: Line integrals of scalar fields with respect to arclength along curves.
• 16.2: Line integrals of vector fields. Arclength, parametric, vector, component, and differential form expressions for line integrals. Work done by forces. Circulation of a vector field around a curve. Flux of a vector field across a curve in the plane.
• 16.3: Conservative vector fields. Equivalence of gradient vector fields and path-independent vector fields. Line integral of a gradient field is the difference of the potential between the endpoints. Zero curl condition for a gradient vector field and determination of the potential from the vector field. Equivalence of gradient vector fields and exact differential forms.
• 16.4: Positively oriented boundary of a region in the plane. Green's theorem in the plane. Circulation-curl and flux-divergence interpretations.

Section 15.8 (on Jacobians and change of variables in multiple integrals) won't be covered on Midterm 2, but it will be covered on the Final.

Some sample midterm problems are here. Some formulas you should know are here.

Solutions to the sample midterm problems are here.

Grade

Grade will based on the midterm and final exams, weighted as follows:

• 30%: Midterm 1
• 30%: Midterm 2
• 40%: Final

Homework will be assigned weekly but will not be collected or graded. Don't expect to pass this course unless you do the homework.

Text

The text is Thomas' Calculus, Early Trancendentals, 14th Edition.

Homework will be assigned from the 14th Edition of the text. There will be no use of MyMathLab or online homework, so all you require for the class is a hard copy or pdf file of the text.

The text is available as a lower-cost, optional e-book through the UC Davis Inclusive Access Program. Click the “Bookshelf” button in the Canvas navigation menu to access your IA Portal and e-book link. You will have 14 days to use the e-book, after which you can choose to opt in or let the access expire. For questions please email the Inclusive Access Help Desk at inclusiveaccess@ucdavis.edu

Syllabus

We will cover most of Chapters 13, 15, and 16 of the text. The main topics are:

• Multiple integrals (Ch 15)
• Vectors (Ch 13.3–13.5)
• Integrals and Vector Fields (Ch 16)

The detailed Department listing of the course syllabus is here.

Homework

Set 1 (Friday, October 4)
Sec 15.1, p. 901: 1, 4, 13, 16, 19, 22, 29, 30, 37, 40
Sec 15.2, p. 909: 3, 6, 11, 15, 19, 28, 31, 33, 36, 47, 59, 62, 83
Sec 15.3, p. 914: 3, 5, 12, 15, 19, 21, 26, 29

Set 2 (Friday, October 11)
Sec 15.4, p. 919: 3, 7, 9, 13, 17, 20, 23, 29, 33, 37, 42
Sec 15.5, p. 929: 3, 9, 13, 17, 20, 21, 23, 37, 41
Sec 15.6, p. 939: Read Sec. 15.6 (This material won't be examined.)

Set 3 (Friday, October 18)
Sec 15.7, p. 949: 6, 7, 8, 11, 17, 21, 23, 26, 29, 33, 36, 37, 39, 43, 49, 53, 55, 56, 65, 71, 81, 86, 87, 104

Set 4 (Friday, October 25)
Sec 15.8, p. 961: 1, 5, 6, 9, 12, 17, 23
Review Ch. 12 and Sec. 13.1–13.2 on vectors from 21C
Sec 13.3, p. 784: 1, 5, 7, 9, 15, 16, 17, 18

Set 5 (Friday, November 1)
Sec 13.4, p. 790: 1, 2, 3, 5, 7, 9, 11, 13, 17, 27
Sec 13.5, p. 797: 1, 5, 3, 7, 9, 11, 13, 26

Set 6 (Friday, November 8)
Sec 16.1, p. 974: 1, 7, 8, 9, 15, 18, 19, 21, 23, 25, 26, 33
Sec 16.2, p. 986: 1, 2, 3, 5, 7, 9, 13, 16, 17, 19, 23, 27, 29, 30, 39, 47, 48, 55, 59

Set 7 (Friday, November 15)
Sec 16.3, p. 998: 1, 4, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 33, 38
Sec 16.4, p. 1010: 1, 3, 5, 9, 10, 11, 15, 21, 25, 26, 27, 31, 37, 43, 45

Set 8 (Friday, December 6)
Sec 16.5, p. 1020: 1, 5, 11, 19, 21, 23, 31, 33
Sec 16.6, p. 1030: 7, 11, 19, 21, 29, 39
Sec 16.7, p. 1043: 5, 7, 12, 13, 15, 19, 21, 27, 31, 34
Sec 16.8, p. 1056: 1, 7, 13, 15, 21, 25, 29, 33, 34, 35, 36