Advanced Calculus II S21

MAA 4212: Advanced Calculus II

University of Florida, Spring 2021

Online and Little Hall 217, MWF3 (9:35–10:25)

Instructor information:

Prof. Vince Vatter

Office: Little Hall 412 (not that it matters)
Office hours (virtual only): Monday 1:30–2:30, Tuesday 11:00–12:00, and by appointment

TA: Ms. Nicole Tuovila
Office hours (virtual only): Tuesday 1:00–2:00


Terence Tao‘s Analysis II, 3th edition.

Course content

From the course catalog: Continues the advanced calculus sequence in limits, differentiation, integration and series.

In addition to mastery of the course content, course objectives include reading, writing, and discovering proofs and constructing proofs and counterexamples in analysis.

Expectations and grading rubric

Work submitted for a grade in this course will be graded in a most rigorous fashion, and thusly, such work should have a good deal of thought and care put into it.

Most of the work required in this course will consist of writing proofs. These proofs will be assigned scores of 0–4 points based on the following guidelines.

0 points.
The work contains no original steps toward a correct solution. This includes work that simply consists of relevant definitions or theorems without interpretation.
1 point.
The work contains some original steps toward a correct solution but does not contain a workable outline of the full solution. This grade is also used if the student has misunderstood the question or made an unwarranted simplifying assumption that makes the problem trivial.
2 points.
The work contains an outline of a correct solution and several steps toward this solution. However, the writing may be unclear, or there may be holes in the argument.
3 points.
The work is resembles a full, complete proof, but it has some serious deficiencies. These may include incomplete sentences, abbreviating words with logical symbols such as those for “for all” or “implies”, imprecise definitions, or overlooking trivial cases. In general this grade is reserved for work which would receive 4 points with minor revision.
4 points.
The work consists of a full, complete proof and is reasonably well written in complete sentences, without logical symbols. There may be minor typos or clumsy writing that could be improved, but no important steps of the solution are omitted or incorrect.


There will be 6 quizzes and 6 homework assignments throughout the semester.

Both quizzes and homeworks will be collected online, via Canvas. Quizzes will be taken remotely, even for those students in the face-to-face section. All quizzes will be open-book and open-note. Late homework will not be accepted except by prior agreement.

All assignments will count equally toward your grade.

Final letter grades will be assigned on a curve, which will be no tougher than the 10-point scale: 90%–100% will be some form of A, 80–90% will be at least some form of B, etc.

If you have a disagreement with the grading of one of your solutions, I ask that you submit a written request for reconsideration within one month.

Grading will be in accordance with the UF policy stated at

Honor code

UF students are bound by The Honor Pledge which states, “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code. On all work submitted for credit by students at the University of Florida, the following pledge is either required or implied: “On my honor, I have neither given nor received unauthorized aid in doing this assignment.” The Honor Code specifies a number of behaviors that are in violation of this code and the possible sanctions. Furthermore, you are obligated to report any condition that facilitates academic misconduct to appropriate personnel. If you have any questions or concerns, please consult with the instructor.


Our class sessions may be audio visually recorded for students in the class to refer back and for enrolled students who are unable to attend live. Students who participate with their camera engaged or utilize a profile image are agreeing to have their video or image recorded. If you are unwilling to consent to have your profile or video image recorded, be sure to keep your camera off and do not use a profile image. Likewise, students who un-mute during class and participate orally are agreeing to have their voices recorded. If you are not willing to consent to have your voice recorded during class, you will need to keep your mute button activated and communicate exclusively using the “chat” feature, which allows students to type questions and comments live. The chat will not be recorded or shared. As in all courses, unauthorized recording and unauthorized sharing of recorded materials is prohibited.

Class attendance

Attendance is strongly encouraged but is not a component of grades.

Requirements for class attendance and make-up exams, assignments, and other work in this course are consistent with university policies that can be found at

Accommodations for students with disabilities

Students with disabilities requesting accommodations should first register with the Disability Resource Center (352-392-8565, by providing appropriate documentation. Once registered, students will receive an accommodation letter which must be presented to the instructor when requesting accommodation. Students with disabilities should follow this procedure as early as possible in the semester.

Online evaluations

Students are expected to provide feedback on the quality of instruction in this course by completing online evaluations at Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at


The official UF policy for filing a complaint about the course may be found here.

Counseling and Wellness Center, 392-1575; and the University Police Department: 392-1111 or 9-1-1 for emergencies.

U Matter, We Care

Your well-being is important to the University of Flo­­rida. The U Matter, We Care initiative is committed to creating a culture of care on our campus by encouraging members of our community to look out for one another and to reach out for help if a member of our community is in need. If you or a friend is in distress, please contact so that the U Matter, We Care Team can reach out to the student in distress. A nighttime and weekend crisis counselor is available by phone at 352-392-1575. The U Matter, We Care Team can help connect students to the many other helping resources available including, but not limited to, Victim Advocates, Housing staff, and the Counseling and Wellness Center. Please remember that asking for help is a sign of strength. In case of emergency, call 9-1-1.

Tentative schedule

Date Topics
1 Monday 1/11 Metric spaces
Section 1.1
2 Wednesday 1/13 Metric spaces
Section 1.1
3 Friday 1/15 Topology of metric spaces
Section 1.2
Monday 1/18 Martin Luther King Jr. Day (no class)
4 Wednesday 1/20 Topology of metric spaces
Section 1.2
Homework 1 due (before class)
5 Friday 1/22 Relative topology
Section 1.3
6 Monday 1/25 Cauchy sequences and completeness
Section 1.4
Wednesday 1/27 Quiz 1 (online, during class time)
7 Friday 1/29 Compactness
Section 1.5
8 Monday 2/1 Compactness, continued
Section 1.5
9 Wednesday 2/3 Compactness, continued more
Section 1.5
Homework 2 due (before class)
10 Friday 2/5 Sequential compactness
Section 1.5
11 Monday 2/8 Sequential compactness, continued
Section 1.5
Wednesday 2/10 Quiz 2 (online, during class time)
12 Friday 2/12 The Heine–Borel theorem
Section 1.5
Continuous functions
Section 2.1
13 Monday 2/15 Continuity and limits
Section 3.1, sort of
14 Wednesday 2/17 Continuity and compactness
Section 2.3
Homework 3 due (before class)
15 Friday 2/19 Uniform continuity and compactness
Section 2.3
16 Monday 2/22 Continuity and connectedness
Section 2.4
Wednesday 2/24 Quiz 3 (online, during class time)
17 Friday 2/26 Pointwise and uniform convergence
Section 3.2
18 Monday 3/1 Uniform convergence and continuity
Section 3.3
19 Wednesday 3/3 The uniform metric
Section 3.4
Homework 4 due (before class)
20 Friday 3/5 Taylor polynomials
21 Monday 3/8 Numerical series and the root test
Wednesday 3/10 Class canceled.
No quiz or homework,
in memory of the Spring Break that would have been
22 Friday 3/12 Formal power series
Section 4.1
23 Monday 3/15 Real analytic functions
Section 4.2
Wednesday 3/17 Quiz 4 (online, during class time)
24 Friday 3/19 Multiplication of power series
Section 4.4
25 Monday 3/22 Linear transformations (review of linear algebra)
Section 6.1
Wednesday 3/24 Recharge Day (no class)
26 Friday 3/26 Normed vector spaces
27 Monday 3/29 Equivalence of norms
28 Wednesday 3/31 The metric space of linear transformations
Homework 5 due (before class)
29 Friday 4/2 The set of invertible matrices
30 Monday 4/5 Derivatives in several variables
Section 6.2
Wednesday 4/7 Quiz 5 (online, during class time)
31 Friday 4/9 Partial and directional derivatives
Section 6.3
32 Monday 4/12 The chain rule
Section 6.4
33 Wednesday 4/14 The contraction mapping theorem
Section 6.6
Homework 6 due (before class)
34 Friday 4/16 Continuous differentiability
35 Monday 4/19 The inverse function theorem
Section 6.7
Wednesday 4/21 Quiz 6 (online, during class time)