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There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, a
by madhadron 8y ago
There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices.
Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).
Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.
Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.
Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.
After that, where you go is going to vary enormously. Based on what you're aiming to do.
- azhenley 8y agoAny suggested reading on rings and lattices? I took a lot of math classes in undergrad but was never exposed to these concepts.
- ducttapecrown 8y agoLattices (the algebraic structure) seem to exist only on Wikipedia in the sense that the only time I've seen lattices every mentioned in my standard undergrad education is once for a proof of the Stone-Weierstrauss theorem. I know there are textbooks on the topic, and probably lots of people who deal with lattices a lot. But my own experience seems to be that wikipedia puts more emphasis on lattices and things like universal algebra than actually happens in math.
- madhadron 8y agoThere is an interesting rant from Gian-Carlo Rota on why lattices are so rarely taught despite being so ubiquitous. I include them based on the amount of use I have gotten out of them over the years. For example, the entire theoretical structure of eventual consistency is "make your merge operation the meet of a semilattice."
- wbl 8y agoHerstein Topics in Algebra or Dummit and Foote.
- mlevental 8y agogoofy suggestion. those are both either grad or senior undergrad books depending on where you are. much better suggestion is A Book of Abstract Algebra by Pinter (really gentle) or A First Course in Abstract Algebra by Fraleigh
- wbl 8y agoI used on in high school and the other my second year of undergrad. They have lots of material but they also have good exercises and explanations.
- dunham 8y agoHerstein was my undergrad book back in the early 90's. I remember really liking it. It's been a very long time, but my recollection was that the groups to rings progression felt a lot more natural to me than the rings to groups progression of some of the other books at the time.
- madhadron 8y agoAny standard abstract algebra textbook will cover rings in gross detail. Dummit and Foote is a much beloved standard, but is also used in graduate classes. I've heard good things about Pinter's book (http://www2.math.umd.edu/%7Ejcohen/402/Pinter%20Algebra.pdf http://www2.math.umd.edu/%7Ejcohen/402/Pinter%20Algebra.pdf). For lattices, there's Birkhoff's book on lattice theory, which is where I learned what I know about it. I haven't spent any time with other books.
- Karrot_Kream 8y agoReally enjoyed Algebra: Category 0. Introduces some basic Category Theory and uses it to develop the standard practice of undergrad algebra. The exercises are medium difficulty (nothing ridiculous like a Knuth or Spivak problem, but nothing breezy either) and there are enough examples in the text of proofs with enough detail to get you working on the exercise problems.
- Koshkin 8y agoDid you mean Aluffi’s Algebra: Chapter 0? What a wonderful book.
- Karrot_Kream 8y agoSorry yes
- gnulinux 8y agoDummit&Foote imho is the best undergrad algebra book.
- oarabbus_ 8y agoMost engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.
- umanwizard 8y agoNeither did most historians, or most apple farmers. What's your point?
- analog31 8y agoFrom what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it. Some of the engineers who attract quantitative work are people who came from outside of the mainstream engineering training, such as scientists and math people.
- you_owe_me 8y agoWhen you refer to engineers, do you mean actual engineers (BSc in Engineering)? or your web designer with jquery skills who calls himself engineer?
- prions 8y agoIn my experience (as a civil then software engineer with a MS in CS in ML/Big Data) "real" engineering is much more prescriptive than software. When you're designing a real-world engineering project, the entire specifications are defined legally (through national, state and local laws) and technically in manuals/books. Many engineering specifications will describe the work done to a T before you even need to think about it i.e. "water main shall be constructed of 12'' coated DIP at depth no less than 2 ft". A lot of the challenges are managerial and logistical. All of this is on purpose. "Traditional" engineering disciplines are more mature and have the constraint of being safe for the general public. There isn't much room at all to creatively deviate from what's already specified. I've found software design to be a lot more technically demanding in regards to designing and building things. There's a lot less precedent, more moving parts and many different ways to do one thing.
- umanwizard 8y agoBasic topology is also important.
- madhadron 8y agoYou're absolutely right, and I kind of quietly slipped it in under "extracting qualitative information about functions" in calculus.