Math

In this section, I have gathered knowledge and resources about a few different areas of math I have learned throughout my time in my Math B.S.. I will assume a knowledge of multivariate calculus of real valued functions (i.e. up to and not including vector calculus). I hope it finds you well and you enjoy yourself!

There is so much more to geometry that just the stuff you learn in high school. The stuff you learned in high school is very important, don't get me wrong, but there are far more advanced concepts that really shake your intuition. For instance, consider the surface of a Pringle chip. Would you say it's flat? No, of course not. So it must be curved, but it's not the same kind of curved that doughnut is. So now you ask yourself, "What's the difference between the 'curvy-ness' of a Pringle and the 'curvy-ness' of a doughnut?" That is a very good question. The answer is part of a field call Non-Euclidean Geometry (Wikipedia, Mathworld). Eucliean Geometry (Wikipedia, Mathworld), the stuff you're familiar with, is extremely important since that's where we live (basically). If you want to practice your Euclidean, and the potential for Non-Euclidean, geometry, I highly suggest the amazing geometry program Geometers Sketchpad.

Have you ever wondered how a spring works? That is, if there were a ball on the end of the spring, then you pulled the spring away from its resting state, how would the ball move? This idea can be modeled by what's called a Differential Equation (Wikipedia, Mathworld), and physically the mathematical model behind it is called the Equation of Motion. There are other kinds of differential equations out there are also model physical systems, like how heat/substance diffuses throughout a medium, or the position of a point on a vibrating string, which are modeled by Partial Differential Equations.

The field of mathematical analysis encompasses a great number of subjects, but the subject that best encompasses the spirit of the core area of mathematics is calculus. The study of differential and integral calculus that one might encounter in freshman/sophomore math classes, up to multivariate calculus and vector analysis, provides a very applied and "This is a thing. This is kind of how it works. Now use it.". In practice, mathematical analysis provides a rigorous way to approach ideas such as sequences, series, derivatives, integrals, and much more, all by the simple notion of comparing things to other things that get arbitrarily small.

This field is also home to subjects such as

• Real Analysis (Wikipedia, Mathworld) studies how real (i.e. not complex) numbers and vectors behave when just in the setting of the standard Euclidean space.
• Measure Theory (Wikipedia, Mathworld) studies spaces that have a "size", and also generalizes the notions of length, area, and volume. A particularly important example is the Lebesgue Measure.
• Functional Analysis (Wikipedia, Mathworld) studies sets of functions and mappings themselves, as well as mappings of functions called Operators, and how they behave in their respective spaces.
• Complex Analysis (Wikipedia, Mathworld) studies the notion of the space of complex numbers. It is in this space and study that the normal rules of real analysis (see above) are a little different (much more structured), and thus allows the existence of phenomena such as "If a function has a single derivative in the complex plane, then it necessarily has infinitely many derivatives in the complex plane", and, surprisingly enough, the standard proof of the Fundamental Theorem of Algebra (Wikipedia, Mathworld). This area is of particular importance to Physics, Engineering, and Applied Mathematics. Here we have a nifty emulation of the Julia set that dynamically updates the constant c in the recurrence relation f(z)=z^2 + c.
• Topology (Wikipedia, Mathworld) studies shapes in the most abstract sense. How topology relates to analysis is really a two-sided coin, because there are aspects of topology that do not really on the fundamental notions of mathematical analysis. For example, much the focus of mathematical analysis is in the context of metric and inner product spaces, whereas there are definitely topological spaces that do not have these structures. If you've ever wondered if you can, without tearing, demorph a coffee cup into a doughnut, and vice-versa, topology says you can.
• Differential Geometry (Wikipedia, Mathworld) studies how a space is curved, and how objects in those spaces change as a result of that curvature from what they "should" be in flat, Euclidean space. This field is the underlying language of the field of physics known as General Relativity.

Like PBS Spacetime, there is a new channel of a similar caliber PBS Infinite Series. Here they explain some rather abstract and high level mathematical concepts that one would normally see in maybe their third, or fourth year in an undergraduate math program, in low-key terms so everybody can enjoy math!

Wolfram isn't just the company that brought you Mathematica, but they also bring us an extraordinary repository of mathematical knowledge format in such a way that makes reading high level mathematica, a breaze, that is Mathworld.