**Lecture room: **B139 Van Vleck
Building

**Lecture time: **Mon-Wed-Fri 11:00-11:50am

Office hours: by appointment

**Lecturer: **Gheorghe Craciun

Office: 405 Van Vleck Hall

E-mail: craciun *at* math *dot* wisc
*dot* edu

**Main textbook: **

*Introduction to Applied Mathematics*, by
Gilbert Strang.

**Other recommended textbooks:**

*1. Advanced Mathematical Methods for Scientists
and Engineers,* by C. M. Bender and S. A. Orszag.

*2. Nonlinear Dynamics And
Chaos: With Applications To Physics, Biology, Chemistry, And Engineering*,
by Steven Strogatz.

*3. Introduction to Applied Mathematics*, by
Laurence Sirovich.

**Course Content: **The course introduces methods
to solve mathematical problems that arise in areas of application such as
physics, engineering, chemistry, biology, and statistics.

**Prerequisites:** Math 319 (ODEs), Math 321 (Vector and
complex analysis), Math 322 (Sturm-Liouville, Fourier Series, intro to PDEs),
Math 340 (Linear Algebra) or equivalent. In addition, Math 521-522
(Mathematical Analysis), and Math 623 (Complex Analysis) are strongly
recommended.

**Grading:** Homework and class participation:
33%, Midterm exam: 33%, Final exam: 33%. *Additional credit can be
obtained for work on class projects, presentations, and group projects. *

**MATLAB help**

Several *MATLAB tutorials* are available here.

A free online textbook “Numerical Computing with MATLAB” by Cleve Moler is available here.

**Review of Linear Algebra**

If you need a detailed review of Linear Algebra you can find Gilbert Strang’s video lectures here.

Also highly recommended: the summary of the book “Linear Algebra Done Right” by Sheldon Axler, available here.

Also highly recommended: the youtube
video channel “*3Blue1Brown” *here.

**Homework 1**

*1. Have a look at these other online resources
for linear algebra: **
*

* **1.1** **The** online textbook** *Linear
Algebra Done Wrong, by Sergei Treil* ** *

* **1.2** **The** wikipedia page *http://en.wikipedia.org/wiki/Linear_algebra

*2. Watch the videos on “Essence of linear
algebra” by 3Blue1Brown here:*

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

*3. Try to solve the following review problems
from the Strang textbook:* **1.5.1, 1.5.3, 1.5.7, 1.5.9, 1.5.11, 1.5.13.**

*Solutions to Homework 1 will not be collected,
but these problems give you an idea of topics that you should already know a
lot about. *

**Homework 2**

*1. Read carefully and in detail the first two
chapters of the** online textbook** *Linear
Algebra Done Wrong, by Sergei Treil* ** *

*2. Select and solve 10 exercises from these two chapters, and submit your solutions by email. Please
include the words “homework” or “hw” in the
subject of your email.*

__Due date: Wednesday Sept 21, just before
class.__

**Homework 3**

*1. Read carefully and in detail the first
Chapters 3-5 of the** online textbook** *Linear
Algebra Done Wrong, by Sergei Treil* ** *

*2. Select and solve about 20 exercises from these
three chapters, and submit your solutions by
email. Alternatively, select any number of exercises (not necessarily
20) to solve and submit, but make sure that you spend at least 6 hours
on this homework assignment. Also, please choose some exercises that are more
challenging for yourself (i.e., don’t just choose the ones that are easier for
you to solve). *

Please don’t forget to include the words

__Due date: Wednesday October 5, just before
class.__

**Homework 4**

*1. Read carefully all the solutions to problems
from the **online textbook** *Linear
Algebra Done Wrong, by Sergei Treil, posted online at https://math.solverer.com/library/sergei_treil/linear_algebra_done_wrong

The same solutions are also *available*** for
download** at https://github.com/HuangJingGitHub/Solution-to-Linear-Algebra-Done-Wrong

*2. Read carefully
Chapters 5-6 of the** online textbook** *Linear
Algebra Done Wrong, by Sergei Treil* ** *

*3. Select and solve about 15 exercises from these
two chapters, and submit your solutions by
email. Alternatively, select any number of exercises (not necessarily
15) to solve and submit, but make sure that you spend at least 5 hours
on this homework assignment.*

*Please don’t forget to include the words “homework”
or “hw” in the subject of your email.*

__Due date: Friday October 14, just before
class.__

__Midterm exam:
Monday October 17,
during class time.__

**Homework 5**

*1. Read carefully the
section on Constrained optimization and Lagrange multipliers in the Strang
textbook, and compare with the Wikipedia page https://en.wikipedia.org/wiki/Lagrange_multiplier*

*Also, have a look at this 3Blue1Brown video
about Fourier series approximations: *

*Also, this site has some illuminating (and
customizable) numerical simulations: **https://www.jezzamon.com/fourier/index.html*

*2. Solve the following problems from the Strang textbook:*

**2.2.1, 2.2.2, 2.2.3(b), 2.2.4, 2.2.5, 2.2.7,
2.2.8, 2.2.11, 2.2.15, 2.2.16.**

*3. Solve the following problems from the Strang textbook:*

**4.1.1, 4.1.2, 4.1.3, 4.1.4, 4.1.5, 4.1.6.**

**Solutions are available here
and here .**

*Please submit your solutions by email, and don’t
forget to include the words “homework” or “hw”
in the subject of your email.*

__Due date: Friday November 4, just
before class (for 50% bonus points), or Monday Nov 7 (for
full credit, but without the 50% bonus points),
__

___________________________________________________

Some additional resources on *distributions:*

1. Introductory notes by Daniel O’Connor on github: https://github.com/danielvoconnor/DistributionNotes/blob/main/distribution_example.pdf

2. Detailed notes on Terry Tao’s webpage: https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/

3.
Fourier series ** completeness**: notes
by Jeremy Orloff at https://physics.purdue.edu/webapps/index.php/course_document/index/phys600/2178/561/20348

___________________________________________________

__Some information about
the final exam:__

*The final exam
will include a Class Project, for which you can choose from the
15 suggestions below.*

*Suggestions for Class
Project topics:*

* *

*1.**Choose your own topic: something that you are interested in and
has connection to applied math. *

*2.**Choose a section from the Strang textbook, or from the Strogatz textbook (or from some other book) that we have
not covered in class, and write a report on that topic, including maybe
solutions to homework problems from that section, and maybe also applications
and examples.*

*3.**Mathematics in the news: find some news stories that have
connections to mathematics, and explain those
connections.*

*4.**Specific applications of mathematics in your field of interest:
engineering, physics, chemistry, biology, etc.*

*5.**Mathematical models of the spread of infectious diseases. *

*6.**Mathematical models in “game theory” and relationships to Linear
Algebra.*

*7.**The diffusion equation.*

*8.**The wave equation.*

*9.**Transport equations.*

*10. Solutions of linear dynamical systems in three or more
dimensions.*

*11. The fast Fourier transform. *

*12. The use of Fourier transforms in signal processing. *

*13. The use of Fourier transforms in image processing.*

*14. Dynamical system models in chemistry or biochemistry.*

*15. Dynamical system models in population dynamics (for example:
competitive exclusion).*

*In general, please also
think about including some use of software (you don’t have to, but it would be
great if you can do so). For example, if your project has any
differential equations, you can include numerical simulations using the
software “pplane”. Otherwise, you can use Matlab, Mathematica, python, Java, or any other software.*

___________________________________________________

**Homework 6**

*1. Have a look at the definition of Fourier
Transform on its Wikipedia webpage https://en.wikipedia.org/wiki/Fourier_transform
(note that it’s a bit different from the definition in our textbook), and also
have a look at this video that describes convolution: https://www.youtube.com/watch?v=851U557j6HE*

*2. For more information about distributions, have
a look at the book “Fourier Analysis and Its Applications” by Anders Vretblad (available online)*

*2. Solve the following problems from the Strang textbook:*

**4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5, 4.3.6, 4.3.7.**

*Please submit your solutions by email, and don’t
forget to include the words “homework” or “hw”
in the subject of your email.*

__Due date: Wednesday November 16, just
before class. __

___________________________________________________

__TAKE-HOME
FINAL EXAM: __

*In addition to the
Class Project described above, solve five of the following 12
problems, and submit your solutions by email (please include “703” or “final”
in the subject of your email):*

*1.
**Explain why the
definition of the Fourer transform for distributions
(i.e., FourierTransform(F)(g) :=
F(FourierTransform(g)) is correct.
*

*2.
**Explain the
mathematics behind the surprising properties of the Borwein
integrals, as described in this youtube video: https://www.youtube.com/watch?v=851U557j6HE
*

*3.
**Find as many
solutions as possible of the matrix equation A ^{3} = I in 3D.
*

*4.
**Find as many solutions
as possible of the matrix equation A ^{3} = I in 2D.
*

*5.
**In general, try to
find all solutions (or as many as possible) for the matrix equation A ^{m } = I in nD.
*

*6.
**Create a list of
“fundamental formulas” for calculating Fourier Transform (such as the formula
for the Fourier Transform of the derivative of a function, etc.) , and prove all or most of them.
*

*7.
**Explain why the
“fundamental formulas” for calculating Fourier Transform also work for
distributions, not only for functions. Are there any exceptions?
*

*8.
**Define “ tempered
distributions” (also known as distributions for the Schwartz test
functions) and explain why the inverse Fourier transform can be calculated
for all tempered distributions.*

*9.
*__Choose your own
problem__*: select some problem that you are
interested in, and is related to some topic we
discussed in this class.*

*10.**
Fill in some of the details of the proof of formula (3) on page 309 of
the Strang textbook, about the “reconstruction” of f(x) from its Fourier
transform.
*

*11.**
For a 1D dynamical system, denote by m, n, p, the number of its stable,
unstable, and half-stable fixed points, respectively. Is there any identity
that the numbers m, n, p must satisfy for all such systems?
*

*12.**
Find a 2D dynamical system that has three limit cycles, of which one is stable
and two are unstable. *