Instructor: Prof. Madnick (email: jesse.madnick@shu.edu)
Office: McQuaid Hall 213
This is the sequel to Real Analysis 1.
Textbooks:
"Calculus on Manifolds" (by Spivak)
"Real Analysis" (by Royden and Fitzpatrick)
Lectures: Muscarelle 09
Mon & Wed: 3:30 pm - 4:45 pm
Office hours: McQuaid Hall 213
Mon: 4:50 pm - 6:00 pm
Tue: 4:00 pm - 6:00 pm
Prerequisites: Solid foundations in real analysis 1, calculus 1, 2, 3, and linear algebra.
Calculus 3: Rapid review of theory (highly recommended)
Calculus 3: Practice problems (also recommended)
If you want even more Calc 3 review, you can check out:
Paul's Online Math Notes (Calc III)
Dr. Madnick's course notes: Vector calculus
(spring 2019 at McMaster University)
Problem Sets: Analysis on Rn
Calculus Set (extra credit) (Due: Wed 1/22) Answers
PSet 0 (Due: Wed 1/22) Solutions
PSet 1 (Due: Wed 1/29) Solutions
PSet 2 (Due: Wed 2/5) Solutions
PSet 3 (Due: Wed 2/12) Solutions
PSet 4 (Due: Wed 2/19) Solutions
PSet 5 (Due: Wed 3/12) Solutions
PSet 6 (Due: Wed 3/19) Solutions
PSet 7 (Due: Fri 3/28) Solutions
Topic 1: Analysis on Rn
Unit 0: Euclidean Space and Multivariable Functions
Unit A: Limits and Derivatives in Rn
A.1-A.4: Limits and Continuity
A.5-A.8: Derivatives
Unit B: The Inverse and Implicit Function Theorems
Unit C: Integrals in Rn
Unit D: Differential Forms and Stokes Theorem
D.1-D.5: Exterior Algebra and Differential Forms
D.6-D.8: Integration and Stokes Theorem
Quiz Information
Date: Mon 2/3
Content: The quiz tests prerequisite knowledge of Calculus 1, 2, 3.
For calculus 1 and 2: See the calculus set
For calculus 3: See the calculus 3 practice problems
Structure: 10 problems.
Problems #1-5: Calculus 1 and 2.
Problems #6-10: Calculus 3.
Midterm 1 Information
Date: Wed 2/26
Content:
Sec 0.3-0.4, Sec A.1-A.8, Unit B
Lectures 1-7
PSets 1, 2, 3, 4
Note:
Sections 0.1-0.2 (and the content of PSet 0) will not be a primary focus of the test. However, that material will be assumed as basic background knowledge.
None of the problems will ask for sketches or drawings.
You may bring 1 page (double-sided) of handwritten notes to the exam. Recommendation: Have the statements of the Inverse and Implicit Function Theorems on hand.
Topics / Structure:
#1. Multiple choice.
#2, #3.
Calculate directional derivatives.
Prove a function is differentiable / not differentiable.
Prove a set is open or closed.
#4, #5.
Use the chain rule.
Use the Inverse Function Theorem.
Use the Implicit Function Theorem.
Extra Credit. One short proof. (Choose one from a list.)
Midterm 2 Information
Date: Wed 4/2
Content:
Units C and D
Lectures 8-14
PSets 5, 6, 7.
Note:
The exam is open book.
There will be a much greater emphasis on PSets 6 and 7 than on PSet 5.
None of the problems will ask for sketches or drawings. However, relevant sketches can earn partial credit.
The volume of a 3-dimensional ball is (4/3)*pi*R^3.
Structure:
#1. Calculate exterior derivative and Hodge star.
#2. Calculate integral of differential form.
#3. Choose 1 problem from a Unit C set of 3
#4. Choose 1 problem from a Unit D set of 3.
#5. Choose 1 problem from either set (Unit C, Unit D) that you didn't solve.
Time management:
- Problems 1 and 2 are meant to be quick (9-10 minutes each).
- Problems 3, 4, 5 are longer (15-20 minutes each).
Final Exam Information
Date: Mon 5/12
Time: 10:10 am - 12:10 pm
Location: Our classroom
Content:
Lectures 15-25
PSets 8, 9, 10, 11
Unit E, Unit F, and Sections G.4-G.5.
Note:
The exam is open book.
Sections G.1-G.3 won't be tested.
Structure: Five problems. Lowest problem dropped.
#1. Choose a problem from a Unit E set.
#2. Choose a problem from a Unit F set.
#3. Choose a problem from a Unit G set.
#4. Choose a problem from any of the three sets.
#5. Choose another problem from any of the three sets.
Why is this course structured in two completely separate parts?
"A year-long course in real-analysis is an essential part of the preparation of any potential mathematician. For the first half of such a course, there is substantial agreement as to what the syllabus should be. Standard topics include: sequences and series, the topology of metric spaces, and the derivative and Riemann integral for functions of a single variable....
There is no such universal agreement as to what the second half of such a course should be... At MIT, we have dealt with the problem by offering two second-term courses in analysis. One of these deals with the derivative and Riemann integral for functions of several variables, followed by a a treatment of differential forms and a proof of Stokes' Theorem for manifolds in euclidean space.... The other deals with the Lebesgue integral in euclidean space and its applications to Fourier analysis." -- Prof. James Munkres