MTWRF 9:10-10:05 at BO (Bolz Hall) 317
| Instructor: Sasha Leibman | office: MW406 office hours: Monday and Thursday 10:10-11am, and by appointment |
| e-mail: leibman.1@osu.edu |
Link to 4181H webpage and lecture notes
Textbook: G.B.Folland Advanced Calculus
Midterm 1 –
solutions.
Topics.
Midterm 2 –
solutions.
Topics.
The final exam
is scheduled for Friday, April 15, 10am-12pm,
and Monday, April 28, 8am-10am
(you can choose either of these two options).
It will be open book and notes, and consist of 6-8 problems
related to some of the following
topics.
Homework:
| Homework 1 – | due by Tuesday, January 14. Solutions |
| Homework 2 – | due by Tuesday, January 21. Solutions |
| Homework 3 – | due by Tuesday, January 28. Solutions |
| Homework 4 – | due by Tuesday, February 4. Solutions |
| Homework 5 – | due by Tuesday, February 18. Solutions |
| Homework 6 – | due by Wednesday, February 26. Solutions |
| Homework 7 – | due by Tuesday, March 4. Solutions |
| Homework 8 – | due by Tuesday, March 18. Solutions |
| Homework 9 – | Not due. Solutions |
| Homework 10 – | due by Tuesday, April 1. Solutions |
| Homework 11 – | due by Tuesday, April 8. Solutions |
| Homework 12 – | due by Tuesday, April 15. Solutions |
Calendar:
| January 6: | Snow | January 7: |
Metric spaces
[Lecture notes 1.1, Textbook 1.1]
Interior, limit, boundary, isolated points [Lecture notes 1.2, Textbook 1.2] |
January 8: | Open and closed sets [Lecture notes 1.2, Textbook 1.2] | January 9: |
Sequences and their limits
[Lecture notes 1.3, Textbook 1.4]
Limits of mappings [Lecture notes 1.4, Textbook 1.3] |
January 10: | Continuous mappings [Lecture notes 1.4, Textbook 1.3] | January 13: | Compact sets [Lecture notes 1.5, Textbook 1.6, 1.5, 1.8] | January 14: | Compact sets [Lecture notes 1.5, Textbook 1.6, 1.5] | January 15: | Connected sets [Lecture notes 1.6, Textbook 1.7] | January 16: |
Path connected sets
[Lecture notes 1.6, Textbook 1.7]
Banach's fixed point theorem [Lecture notes 1.7 |
January 17: | Vector spaces, linear mappings, matrices [Lecture notes 2.1-2, Textbook Appendix A] | January 21: | Dual space, bilinear mappings, norms [Lecture notes 2.3-5, Textbook Appendix A] | January 22: | Norms, scalar multiplication [Lecture notes 2.5-6, Textbook Appendix A] | January 23: |
Determinant
[Lecture notes 2.7, Textbook Appendix A]
Multidimensional differentiation [Lecture notes 3.1, Textbook 2.2, 2.10] |
January 24: | Differentiation [Lecture notes 3.1, Textbook 2.10] | January 27: | Directional and partial derivatives, the Jacobian matrix [Lecture notes 3.2, Textbook 2.2, 2.10] | January 28: |
Continuous partial derivatives imply differentiation
[Lecture notes 3.2, Textbook 2.3]
The chain rule [Lecture notes 3.2, Textbook 2.3] |
January 29: |
The mean value theorem
[Lecture notes 3.3, Textbook 2.4]
Extreme and critical points [Lecture notes 3.4, Textbook 2.8, 2.9] |
January 30: |
Extreme and critical points, Lagrange's multipliers
[Lecture notes 3.4, Textbook 2.9]
The second derivative [Lecture notes 3.5, Textbook 2.6] |
January 31: |
Commutation of directional derivatives
[Lecture notes 3.5, Textbook 2.6]
The second derivative of a composition of two functions [Lecture notes 3.5, Textbook 2.6] |
February 3: |
Behaviour of a function near a critical point via its second derivative
[Lecture notes 3.5, Textbook 2.6]
Higher order derivatives [Lecture notes 3.6, Textbook 2.6] |
February 4: | Multivariable Taylor polynomials [Lecture notes 3.6, Textbook 2.6] | February 5: | Review. | February 6: | Midterm 1. | February 7: |
Differential operators.
Gradient is not a vector. Variational problems [Lecture notes 3.7] |
February 10: |
Variational problems
[Lecture notes 3.7]
Homeomorphisms and diffeomorphisms [Lecture notes 4.1, Textbook 3.4] |
February 11: | The inverse mapping theorem [Lecture notes 4.1, Textbook 3.4] | February 12: | Manifolds [Lecture notes 4.2, Textbook 3.3] | February 13: |
Smooth manifolds
[Lecture notes 4.2, Textbook 3.3]
Orientable manifolds [Lecture notes 4.2, Textbook 3.3] |
February 14: | Exercises from Textbook, Sections 3.3, 3.4 | February 17: | Manifolds with borders [Lecture notes 4.2, Textbook 3.3] | February 18: | The implicit mapping theorem [Lecture notes 4.3, Textbook 3.1] | February 19: | Applications of the implicit mapping theorem and examples [Lecture notes 4.3, Textbook 3.2, 3.3] | February 20: | The structure of mappings of constant rank [Lecture notes 4.4, Textbook 3.5] | February 21: |
Functionally dependent functions
[Lecture notes 4.4, Textbook 3.5]
Exercises from Textbook, Chapter 3 |
February 24: | Integration [Lecture notes 5.1, Textbook 4.2] | February 25: | Integration [Lecture notes 5.1, Textbook 4.2] | February 26: |
Integration over Jordan measurable sets
[Lecture notes 5.1, Textbook 4.2]
Properties of integral [Lecture notes 5.2, Textbook 4.2] |
February 27: | Properties of integral [Lecture notes 5.2, Textbook 4.2] | February 28: | Sets of zero content and volume [Lecture notes 5.3, Textbook 4.2] | March 3: |
The volume under the graph of a function
[Lecture notes 5.3, Textbook 4.2, exercise 2]
Fubini's theorem [Lecture notes 5.4, Textbook 4.3] |
March 4: | Fubini's theorem, Cavalieri's principle [Lecture notes 5.4, Textbook 4.3] | March 5: | Change of volumes under a linear transformation [Lecture notes 5.5, Textbook 4.4] | March 6: | Change of variables in an integral [Lecture notes 5.5, Textbook 4.4] | March 7: | Change of variables in an integral [Lecture notes 5.5, Textbook 4.4] | March 17: | Improper integrals [Lecture notes 5.6, Textbook 4.6-7] | March 18: | Improper integrals [Lecture notes 5.6, Textbook 4.6-7] | March 19: | Functions defined by integration [Lecture notes 5.7, Textbook 4.5, 7.5-6] | March 20: | Functions defined by integrals [Lecture notes 5.7, Textbook 4.5, 7.5-6] | March 21: |
The gamma function
[Lecture notes 5.7, Textbook 7.6]
Examples of improper integrals. |
March 24: | Exercises from the textbook. | March 25: | Midterm 2. | March 26: | Transformation of the n-dimensional measure under linear mappings [Lecture notes 6.1 | March 27: | Exterior multiplication [Lecture notes 6.1, Textbook page 270] | March 28: | Integration over n-dimensional sets [Lecture notes 6.2, Textbook 5.1, 5.3] | March 31: |
Integration over a graph of a function and over a surface of revolution
[Lecture notes 6.2, Textbook 5.3]
Partition of unity [Lecture notes 6.2, Textbook B.7] |
April 1: |
Integration of vector fields
[Lecture notes 6.3, Textbook 5.2, 5.3]
Gradien, divergence, and curl [Lecture notes 6.3, Textbook 5.4] |
April 2: |
Relations between gradient, divergence, and curl
[Lecture notes 6.3, Textbook 5.4]
Green's, Gauss's, and Stokes's theorems [Lecture notes 6.3, Textbook 5.2, 5.5, 5.7] |
April 3: | Examples for Green's, Gauss's, and Stokes's theorems, exercises [Lecture notes 6.3, Textbook 5.2, 5.5, 5.7] | April 4: | Conservative vector fields [Lecture notes 6.4, Textbook 5.8] | April 7: | Differential forms [Lecture notes 6.5, Textbook 5.9] | April 8: | Differentiation of differential forms [Lecture notes 6.5, Textbook 5.9] | April 9: | Pullback of differential forms [Lecture notes 6.5, Textbook 5.9] | April 10: | Integration of differential forms [Lecture notes 6.6, Textbook 5.9] | April 11: | The general Stokes theorem [Lecture notes 6.7, Textbook 5.9] | April 14: | The general Stokes theorem [Lecture notes 6.7, Textbook 5.9] | April 15: | The general Stokes theorem [Lecture notes 6.7, Textbook 5.9] | April 16: |
Poincare's lemma
[Lecture notes 6.8]
Series in normed vector spaces [Lecture notes 6.7, Textbook 7.1, 7.2] |
April 17: | Functional sequences and series [Lecture notes 6.7, Textbook 7.1, 7.2] | April 18: |
Review.
Integral operators, convolution [Lecture notes 5.7] |
April 21: | Functions, defined by integration on a parameter [Lecture notes 5.7, Textbook 4.5, 7.5-6] |