Applications of Math: Examples & Fields

[MathWarrior]

This is probably a commonly asked question but I felt id ask it. If your studying math and you get really good at the good basics of math, algebra, linear algebra, real analysis, differential equations, PDE's, calculus, trigonometry, etc. etc. I can see a lot of small examples of where it is used. But are there any areas that really require you to understand the whole picture?

For instance, if you were going to build a robot you would need engineering skills etc., there would be parts where you would need math and I can see it being used there, but its clearly not the tool for every job? Or is it?

I realize there's a great many applications for math, but they seem rather large and diverse. Does anyone know of some specific examples or examples of math being used enormously in particular fields.

I just am wondering if someone can expand on more areas where and how its used basically.

 

[chiro]

Math is basically a way to generalize things and to use the framework of generalization and analysis to find out something.

So expanding on the above you take some details you know and some assumptions that you are making, then depending on what you want to find use some mathematics and depending on what you find out, make a decision based on that.

Every area of math creates a framework for generalization in a very specific way. Statistics deals with general ways to analyze data under the framework of uncertainty (think randomness).

Calculus creates a framework of analyzing general functions of one and many variable and has uses such as finding different measures and finding optimal solutions to systems with many variables as well as constraints. Remember its a framework that works with general functions and not specific ones which means that if the proof for the general function works, then any function can be used in that framework.

Linear algebra deals with linear systems. This also relates to calculus because derivatives in many dimensions are themselves linear objects. Many problems can be expressed using matrices and all these problems generally use the same technique of analyzing linear systems to find a solution.

On the topic of linear algebra many vector spaces also have inner product spaces of which orthogonality provides a framework for decomposition of systems. The infinite dimensional version of this allows us to look at decomposing functions and signals into "vectors" on infinite dimensional spaces that have a specific purpose.

For example Fourier analysis let's us take a signal and get rid of higher frequency contributions. We can do this to get rid of unwanted noise which is used in many many applications from analyzing audio to analyzing data in a communication context.

Basically when you think of math, you should think about thinking of ways to look at general things. When you did math in high school you probably learned about specific things like finding areas of objects with straight lines and you probably looked at formula after formula describing specific things.

Pure math is the opposite of this. We start out with something that is really broad and we investigate it and try to make sense of it in a way that we can analyze it effectively.

The applied scientists and engineers take this framework and apply it to specific situations.

Basically anything that has to be modeled, analyzed, and made sense of will use math because it provides a generic framework for doing just that.