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Archive for August 31st, 2022

Base Matter Man

Aug 12, 2022

All the math you need for a Física Degree

When you study physics you use a lot of math. This puts people off from studying it, but it can be far less frightening when you know what you’re up against.

I’ve just completed my degree, and I want to talk about it. So, here’s a brief overview of all the math classes I took.

Precalculus

Precalculus should not exist. It is a testament to how poorly we teach math in American high schools. All it is, is a hodgepodge of concepts from Algebra, Geometry, and Trigonometry, which happen to come up often in Calculus.

I say it shouldn’t exist, because we should learn these concepts in our ordinary algebra/geometry classes, but we don’t. That being said, there is nothing to fear here.

There’s no real prerequisites to learning it. You do need to learn it though. You don’t need to be a pro at calculus to get a physics degree, but you do need to be able to do trigonometry in your sleep.

Math is a language, and algebra is its grammar.

You have to learn the basics first.

Once that’s out of the way, you learn Calculus. You do this in 3 stages, whose naming convention is terrible. Cálculo I is differential Calculus. Calculus II is Integral Calculus. Calculus III is three-dimensional calculus.

Calculus I (Derivatives)

In calculus I you learn about Limits, that is, approaching something forever without ever actually touching it. This is a far more profound idea than you will ever realize and many people have dedicated their entire lives to understanding it. It is this base idea, from which the rest of calculus arise.

After you get a basic handle on limits, you learn derivatives. A derivative is a measure of how something changes. The derivative of position is velocity, and the derivative of velocity is acceleration. this is why calculus is sometimes called the “study of change”. (And what could be the derivative of acceleration? any third derivative in physics?)

Nature is in a constant state of flux, physics is partially the study of this flux, so understanding change is imperative.

Calculus II (Integrals)

In arithmetic, there are basic operations: addition, subtraction, multiplication, and division. In calculus the two fundamental operations are Derivatives and Integrals.

You get derivatives in Cal I and you get Integrals in Cal II.

Loosely speaking an integral is the area under a curve, which might seem unimportant but you can then manipulate integrals to find the area of any shape.

In classical geometry we can find the area of circles, triangles, and rectangles. In calculus, you can find the exact area of any shape imaginable. It’s hard to explain how significant that is, but all of physics moving forward depends on it.

Integrals are considerably harder to solve than Derivatives, so this course will be taught like a tool bag of techniques for solving different integrals.

You’ll also discuss infinite sequences and series, that is infinite lists of numbers following some pattern and infinite summations of those lists of numbers.

Sequences and Series have interesting philosophical implication and aren’t too hard to understand, but i would rank this as the hardest math course I had to take while in college.

It gets easier after this.

Calculus III (3D Calculus)

Once you master Derivatives and Integrals, you are taught to generalize those ideas to three dimensions. This involves lots of new “analytic Geometry” and the most beautiful theorems.

Its a difficult class but in my opinion it is the most beautiful. Everything you learn builds on what came before, and it all leads up to two incredibly important theorems: Stoke’s theorem and the Divergence theorem, which you learn at the very end of the course. To understand them, you need to understand Everything that came before. Calculus II is an unorganized hodgepodge of techniques like Precalculus, whereas everything in Calculus III is extremely methodically laid out. It all culminates in Stokes and Divergence.You need calculus III when analyze any kind of a field. First and Foremost you will need it to study the magnetic and electric fields in ElectroDynamics, but also the gravitational fields in Classical Mechanics, as well as water in fluid mechanics, and it even has applications in General relativity. I was proud when I finished this course. I had to retake it a few times. I remember thinking “I’m done, I did it.”Linear Algebra (Multidimensional Algebra)It’s hard to say what linear algebra is, there’s a few ways to think about it. Officially it’s the study of linear equations, but that doesn’t do it justice. I’ve also heard it described as the study of vector fields, but I passed it without knowing what a vector field was. It’s definitely an entirely different thing from calculus and despite what your advisor tells you, calculus is not a prerequisite. Here you don’t deal with Integrals and Derivatives, but with matrices and vectors. two dimensional blocks of numbers and one dimensional Lists of numbers.It is basically algebra, but generalized to higher dimensions (as many as you like), with the restriction that we only work with equations that make straight lines when you graph them. It’s relatively easy, but extremely tedious, a single problem can take upwards of an hour, which is why we often program computers to help us. A smart high schooler could learn the basic rules easily. The only reason they don’t is because we are taught not too. Finally it is incredibly powerful. You will use it everywhere. It is the language of Quantum mechanics, and a prerequisite for Tensor Calculus, the math of general relativity. That’s getting ahead of myself, It has applications in everything to follow. In the most mundane of tasks. Everything. Absolutely Everything, somehow relates to linear algebra.Ordinary Differential Equations (single dimensional)A differential equation is an equation that contains derivatives (like from Calculus I). the best way to explain it is to compare it to an algebra equation. You solve an Algebra equation for x, but you solve a differential equation for an algebra equation.It isn’t an exaggeration to say that nearly all of physics is differential equations so this is one of the most important classes you will take. Newton’s second law for instance, Force = Mass x Acceleration is a differential equation. Acceleration is the Second derivative of Position. By solving, we can determine an objects position based on the Forces applied to it.Partial Differential Equations (Multi dimensional )“Partial” makes it sound easier, but that is certainly not the case. This Course is differential equations multiple dimensions. It is extremely difficult, but you can and will get through it. Here you will learn how to solve the heat equation and the wave equation.I had to take this class more than once. At a certain point I realized, PDE is fundamentally about linear algebra, not Differential Equations. A PDE solution is a generalization of a kind of Vector. Once you start seeing it like that, it starts to fall into place.PDE is important for Waves, any kind of changing phenomenon in multiple dimensions, and it is the language of quantum mechanics.That covers all the required math, but most physics students will take a few extra, just for kicks. I mean, if you don’t like math, you really shouldn’t be getting this degree.Foundations of Matemáticasyou might ask yourself, “why does calculus work?” or “where does the quadratic formula come from?”. They are derived from more fundamental truths, which arise from even more fundamental truths, and this goes on for some time…Around the start of the 20th century, mathematicians became very concerned with grounding all of math in a set of undeniable axioms, which could be used to reproduce all of mathematics. It turns out this is an incredibly difficult and controversial task, but one worth studying.Basically foundations is a class on proofs. You will learn set theory and have many profound philosophical discussions about what a number is. It’s a great course to take if you plan on studying pure math, because it gives you an opportunity to learn how to prove something before your feet are put to the fire in a class like Real Analysis. It is also great if, like me, you plan on studying mathematical philosophy later on in grad school. Much mathematical philosophy deals with the paradoxes that arise when you study foundations.Numerical Methodsmost math classes teach what can be thought of as “Analytical methods”, that is, you learn how to solve it for an exact solution. That’s great! That’s what we want! unfortunately it’s often impossible. We do not understand how to solve the Vast majority of differential Equations, for example. If you can find an analytic solution to the Navier stokes equation, you will be rewarded a million dollars and you will be remembered forever.This was a revelation for me. I think in high school I assumed that, we as humanity, just sort of had math figured out. I also didn’t realize just how much math there was. Not the case at all. If you have an álgebra equation, even if it’s very hard, somebody can solve it, but there are many differential equations that NO ONE knows how to solve. Some people devote their lives to understanding them.A numerical method, is kind of like trial and error. You guess and check a solution over and over again, usually with some rule that makes your next guess slightly better, and you keep doing that until you approximate a solution. It’s a dry brute force sort of course, with plenty of help from computers. It’s the sort of work we used to get slaves to do, as its not very creative and involves a lot of doing the same thing over and over again until it’s good enough. Euler’s method and Runga-kutta are two ways of solving a differential equation without actually solving it. Dull, but important to learn.Statistics and ProbabilityYou use statistics for everything in physics, especially for lab work, but they never make you take an official class on it so lots of physics students are very confused. Probability is extremely important for quantum mechanics as well as Statistical Mechanics (microscopic thermodynamics). All of statistical mechanics is just basic probability theory. It’s pretty dry, but good to know, and definitely the easiest math class you’ll take. My university taught it in the english literature building for some reason.Complex VariablesComplex variables is essentially Calculus with imaginary numbers. It is Divine. It is pure yet applied. The last math class that I took and I am so happy that I did. It isn’t too hard, but it has a myriad of applications in many branches of physics as well as many in electrical engineering. It also helps you get a feel for imaginary numbers, which show up anywhere there are waves, especially in quantum mechanics. The whole course works towards the residue theorem, which is a method of integrating in a loop on the complex plane. I can’t convey how beautiful it is here. You have to experience it first hand. My favorite math class by far, and the last one I took.Those were all the math classes I took. Is that all the math you need for physics? God no. There is so much more math than you will ever realize, and the Type of physics you specialize in determines the type of math you need to study. I’m currently trying to learn Tensor Calculus, which is kind of like a cross between Cal III and Linear Algebra and is the language of General Relativity. Education is a continuing process, these suffice for an undergraduate physics degree. The core courses you need for physics:Calculus, Linear Algebra, Differential Equations


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