## My Journey Through Mathematics

A photo of me and my library of math books that I studied to walk with a Bachelor’s of Science in Applied Mathematics.

As of April 22, 2011, I have a Bachelors of Science degree in Mathematics, with an emphasis in applied math, and a minor in Computer Science, with an emphasis towards software development. While going through my college career in math, I wanted to take as many diverse electives as I could, while still maintaining the requirements for graduation in a timely manner. What resulted was not only study a great deal of applied mathematics, but also reaching the requirements for a an emphasis in general mathematics theory as well (even though the university won’t give me a second emphasis).

Here are the classes I studied, and the topics discussed in each class:

Precalculus:

1. Graphs
2. Functions and Their Graphs
3. Polynomial and Rational Functions
4. Exponential and Logarithmic Functions
5. Trigonometric Functions
6. Analytic Trigonometry
7. Applications of Trigonometric Functions
8. Polar Coordinates and Vectors
9. Analytic Geometry
10. Systems of Equations and Inequalities
11. Sequences, Mathematical Induction and the Binomial Theorem
12. Counting and Probability

Single Variable Calculus I:

1. Function and Models
2. Limits and Rates of Change
3. Derivatives
4. Applications of Differentiation
5. Integrals
6. Applications of Integration

Single Variable Calculus II:

1. Inverse Functions
2. Techniques of Integration
3. Further Applications of Integration
4. Differential Equations
5. Parametric Equations and Polar Coordinates
6. Infinite Sequences and Series

Multivariate Calculus:

1. Vectors and the Geometry of Space
2. Vector Functions
3. Partial Derivatives
4. Multiple Integrals
5. Vector Calculus
6. Second-Order Differential Equations

Discrete Mathematics:

1. Logic, Sets and Functions
2. Algorithms, Integers and Matrices
3. Mathematical Reasoning
4. Counting
6. Relations
7. Graphs
8. Trees
9. Boolean Algebra
10. Modeling Computation

Ordinary Differential Equations:

1. Classifications
2. First Order
3. Second Order
4. Higher Order
5. Series Solutions of Second Order
6. The Laplace Transform

Elementary Linear Algebra:

1. Vectors
2. Solving Linear Equations
3. Vector Spaces and Subspaces
4. Orthogonality
5. Determinants
6. Eigenvalues and Eigenvectors
7. Linear Transformations
8. Applications of Linear Algebra

Probability and Statistics I:

1. Historical Summary
2. Probability
3. Random Variables
4. Special Distributions
5. Estimation

Dynamical Systems:

1. Differential Equations
2. Planar Systems
3. Interacting Species
4. Limit Cycles
5. Hamiltonian Systems, Lyapunov Functions and Stability
6. Bifurcation Theory
7. Three-Dimensional Autonomous Systems and Chaos

Mathematical Modeling:

1. One Variable Optimization
2. Multivariable Optimization
3. Computational Methods for Optimization
4. Introduction to Dynamic Models
5. Analysis of Dynamic Models
6. Simulation of Dynamic Models
7. Probability Models
8. Stochastic Models

Elementary Topology:

1. Set Theory and Logic
2. Topological Spaces and Continuous Functions
3. Connectedness and Compactness
4. Countability and Separation Axioms

Numerical Analysis I:

1. Preliminaries
2. Solutions of Equations in One Variable
3. Interpolation and Polynomial Approximation
4. Numerical Differentiation and Integration
5. Initial-Value Problems for Ordinary Differential Equations
6. Direct Methods for Solving Linear Systems

Number Theory:

1. The Integers
2. Integer Representations and Operations
3. Primes and Greatest Common Divisors
4. Congruences
5. Applications of Congruences
6. Special Congruences
7. Multiplicative Functions
8. Cryptology
9. Primitive Roots

Elementary Real Analysis I:

1. Properties of the Real Numbers
2. Sequences
3. Sets of Real Numbers
4. Continuous Functions
5. Differentiation
6. Integration

Elementary Real Analysis II:

1. Infinite Sums
2. Dense Sets, Oscillation and Continuity on Sets
3. Sequences and Series of Functions
4. Power Series
5. Measure Theory
6. Lebesgue Integration

Even though I’ve graduated with a math degree, I think I would like to attend school further to pick up Modern Algebra I and II, Complex Analysis, Matrix Theory and maybe one or two more before attempting the math GRE and applying for graduate school. There are many more math classes in the undergraduate program to take, such as History of Mathematics, Boundary Value Problems, Partial Differential Equations, Probability and Statistics II, Numerical Analysis II, Enumeration, Euclidean and Non-Euclidean Geometry and Graph Theory.

I’ve found a website that aggregates free electronic books in various formats, such as PDF. Tons and tons of mathematics books are referenced to on the site, many of which I’ve downloaded and begun reading. The site is: http://e-booksdirectory.com/listing.php?category=3. I hope I never lose my passion for

1. Jef Spaleta using on | April 22, 2011 at 4:02 pm | Permalink

I’m tempted into getting into “mine is bigger than yours” competition with regard to math text books. But I will refrain.

A suggestion, if you want to really followup on the non-discrete side of applied math, audit some physics or engineering classes, especially anything that deals with fluid-like systems or fields. BVPs and PDEs are the lingua franca in those spaces and the best way to learn the language is to use them in conversation. I said this as a Math/Physics double major undergraduate.

-jef

2. Johan using on | April 22, 2011 at 9:24 pm | Permalink

I’d definitely recommend taking a complex analysis course. If you can find a course in basic differential geometry, that’s also fun.

3. using on | April 23, 2011 at 8:12 am | Permalink

Wow, you’re really lucky to have been able to learn so much already! I hope to one day take some time off and learn some maths.

4. using on | April 24, 2011 at 6:50 am | Permalink

@Jef Spaleta- This post wasn’t meant to be a “mine is bigger than yours” competition. It was meant to be interesting, in that I had the opportunity to study a great deal of mathematics during my undergraduate career, and I wished to share what I learned.

@Johan- Complex analysis was offered during a time that I could take it with modern algebra, however, they adjusted the days it will be taught, and I can no longer take the course. Differential geometry isn’t offered at my school, although I do plan on taking differential topology in graduate school.

@Jonathan Carter- Maths ftw!

5. tyg using on | April 24, 2011 at 11:08 am | Permalink

Check this site out for a great list of open content math books:

http://linear.ups.edu/curriculum.html

6. Jef Spaleta using on | April 25, 2011 at 5:39 pm | Permalink

@Aaron,

It was meant as a bit of self-deprecating snark. Apologies if it came off as some sort of comment on your scholarly career as it was not intended to be. Though saying it was not what I intended is no excuse for causing offense, so apologies once again.

However, the comment about seeking out cross-disciplinary experience with PDEs and BVPs was meant to be taken seriously.

-jef

7. using on | April 26, 2011 at 6:39 am | Permalink

@Jef Spaleta- No offense taken, I just wanted to be clear for future readers that this isn’t a bragging post, but meant to be an informative one. Many people who aren’t involved with mathematics beyond high school, or what is required for their degree, think Calculus is the highest level of math one could learn. Hopefully, this post was meant to be enlightening to those people that there is a great deal of mathematics beyond calculus, and even beyond undergraduate school (even though that’s not explicit here). Anyway, no offense taken, so no need to apologize.

My school only offers BVPs and PDEs every other year (BVP in the fall, and PDE in the spring). It was offered this past year, so it will be two more years before offered again, and hopefully, I’ll be pursuing my masters degree in mathematics at that point.

When I was looking at the courses that I wanted to take for my undergraduate degree, I decided that I would align all of the applied mathematics requirements as electives for the general math emphasis. This turned out to be a smart move (a story I won’t go into here), as I needed to switch my emphasis from general math to applied math. BVPs and PDEs weren’t requirements for the general track of the applied emphasis, only for the physics track. Further, I wanted to take as many diverse courses as possible during my studies, while still maintaining the requirements for graduation, and it seemed to me (I am likely wrong), that BVPs and PDEs are just extensions of ODEs. Because I have already taken ODEs, I wanted to see what else was out there that was completely unrelated to what I had already studied.

Come this fall, even though I am now graduated, I will be taking the Modern Algebra track, so I can still say that I have met all the requirements for both the general and applied requirements, even if the school won’t give me two degrees.

8. Jef Spaleta using on | April 26, 2011 at 10:36 am | Permalink

@Aaron
Well everything is just an extension of set theory. We should all just be taught that in 3rd grade and be done with math forever.

In terms of math bag of tricks, I found Graph Theory’s heavy reliance on inductive proof very different than the other math courses in a get my brain thinking out of the box standpoint. Graph theory is very fun to play with as a napkin proof hobby. Very doodle friendly.

Linear Programming and Non-linear Programming were interesting as well in terms of getting a better understanding the problem space of optimization in high dimensional spaces. Badly named subjects to be sure.They should have been called Linear and Non-Linear Optimizations.

One of the best math classes I had was an Asymptotic class which was very very interesting. Graduate level class for physics actually. Practical application of taking limits when working with differential equations in order to order terms and shake out dominate solutions out of horribly non-linear situations. Very useful. A class full of bags of tricks that I wouldn’t have minded seeing in a math rigorous approach.

-jef

9. using on | April 26, 2011 at 1:35 pm | Permalink

@Jef Spaleta- Fair enough. All math is an extension of set theory and the axioms based on them. But, I think it’s fair to say that there are various branches of mathematics that are drastically different than others. For example, Real Analysis is drastically different from Number Theory which is turn is drastically different from Numerical Analysis, and so on. It’s fair to say, that while they all have a parent root, they can be quite different. I was hoping to learn these various differences.

Which brings to my mind, and maybe I should put this in a separate post altogether, but here are the various branches of mathematics that I’m familiar with- hopefully, I can learn something from each:

• Calculus and Analysis
• Geometry and Topology
• Algebras
• Set Theory and Logic
• Probability, Statistics and Combinatorics

In regards to the various topics you presented, I would LOVE to learn Graph Theory. In fact, it’s offered this fall semester, and if it aligns on the same days as Modern Algebra, I’m all over that like white on rice.

10. Jef Spaleta using on | April 26, 2011 at 2:40 pm | Permalink

I’m not sure where you put graph theory in that grouping. It’s just …different. I wouldn’t necessarily call it an algebra.

-jef

11. using on | July 7, 2011 at 6:24 am | Permalink

mast nahi hai yaar

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