pure math probably has the "hardest" courses in terms of a lot of proof heavy and very abstract classes. but a lot of classes from other disciplines can also be very difficult, e.g. control theory and signal processing in engineering can be tough and also a few classes in stats like stochastics, probably some econometrics classes.
first year math classes that a lot of majors have to take (e.g. math 1a, math 2, similar) won't really touch the surface of anything difficult at all. usually, these classes just continue on from high school, more vectors and linear algebra, more calculus, probably multivariable calculus and more statistics. from there most unis will branch out the classes into pure math and applied math (+ stats) with some overlap. here's a rundown of the main classes (it differs a bit from uni to unit though):
core to all the majors:
- first year stuff: usually this means introductory calculus and linear algebra. for calculus you basically go through the same thing as in high school except for in a bit more detail, e.g. more infinite series and sequences, more methods to solve differential equations, more integration techniques, etc. then you will probably go into multivariable calculus which is basically more of the same, except now you deal with functions of more than one variable, nothing too extreme. linear algebra then generalises stuff about vectors and simultaneous equations, so you learn about these things like matrices that allow you to more easily manipulate simulatenous equations instead of writing (1), (2), (1) + (2) and so on, and also some more vector operations. basically, this is just revealing the next layer of computational stuff from high school
- linear algebra: this is the second run through of this topic, except this time you prove all the statements. for example in first year you learn about how to solve linear systems of equations with any number of variables, but in this class, you prove that all the things you did actually work, e.g. what criteria tell us when there's an unique system to the system of equations, what actually are vectors (they're not just arrows, for example the continuous functions can be vectors too), etc.
linear algebra is super core because almost every other subfield of math uses it, e.g. if you want to solve more complicated differential equations, turns out in general you need linear algebra, for stats it's used in plenty of models, image compression etc. for a lot of people this might also be the first time that you get introduced to more abstract proofs, so here you prove about ANY system of linear equations, not just a particular type of them, unlike the very specific statements in high school
pure math: usually this just refers to "proof based classes" although this is also true for the other majors... but these are typically the most theoretical type of classes.
- real analysis: this basically involves working out how the real numbers work, and how you can use these properties to study calculus properly. e.g. here you define what a limit is properly, derive all the properties of derivatives using definitions, give criteria for general sequences of numbers to converge, define an integral properly and so on. also, you learn about how calculus isn't really a complete theory: for example there are functions that are continuous everywhere, but differentiable nowhere, and the integral you learn about in high school fails to work on a lot of simple functions (for example, try to integrate the function that's 1 at the rational numbers, but 0 at the irrationals...). this is usually perceived as a "trial of fire" class as it can be very difficult, but if you're interested in learning about why calculus actually works, it's a really fun class
- abstract algebra: this class is about what "algebra" means in the modern sense, which is essentially describing some type of structure that is shared between a lot of different objects. for example, a rubix cube, the integers with the operation of addition, and the set of all permutations of the numbers from 1 to n with function composition all have the same structure. so you learn about how to describe these more general setups, and from describing the general structure, you can describe a lot of different things at the same time
- some electives: from here it branches out. you can do more analysis which lets you study calculus in more abstract settings (e.g. complex numbers, or functions themselves - not the variables, e.g. how can you define the "limit" of one function approaching another?), more abstract algebra where you learn about more complex structures, differential equations in more than one variable, how to describe weirder geometries etc
applied math: as the name suggests this is usually more focused on giving applications ontop of abstract notions
- computational maths: here you learn about a lot of approximation methods. it turns out that in general, differential equations don't have a nice closed form solutions, so how can you approximate a solution rather than solving it directly? similarly you can approximate the solutions to a large degree polynomial, or solve an integral approximately using a lot of different methods, which is all of what this class is about. of course because it involves differential equations there's plenty of applications to basically any field you mention
- differential equations: as it sounds you solve more complicated differential equations, which usually involves determining which "class" of differential equation you're dealing with, then applying specific methods to solve them. again obviously plenty of applications
- optimisation: this class is about more general optimisation problems. in high school you already know about how to find a minimum of one function, but what about if you want to find a minimum solution when you have several functions with many constraints? for example say you wanted to minimise the number of calories you eat and maximise the number of calories you burn, but you have constraints on the amount of carbs, protein etc you need to intake in a day. obviously that's a simple example, but in general you can have many many variables and many many constraints, and in this class you'd learn about how to figure out when you have a solution and how to approximate these solutions. this is super useful for stuff like logistics, e.g. scheduling flights or determining the best routes for trucks to drive and things like that
again you can go into further detail into a lot of things, usually people pick stuff like fluid mechanics or biology adjacent stuff where they learn about more specific models rather than learning about more methods as at this point you're pretty well equipped to tackle more real world problems
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