Math

Keith Devlin's Mathematical Thinking looks to be a good introduction to the topic.

Many recommend Knuth's book: Concrete Mathematics, an Introduction to Computer Sciences

Calculus: rbs0 recommends Calculus and Analytic Geometry by Thomas. Others recommend Calculus an Intuitive and Physical Approach and What Is Mathematics?: An Elementary Approach to Ideas and Methods (recommended by Einstein)

• Lagrange Multipliers (with application in Calculus of Variations)

Alex recommends reading any of George Polya's books, with How To Solve It and Induction And Analogy In Mathematics and Patterns of Plausible Inference being a few. Also, doing math exercises, he recommends Art of Problem Solving website.

• Math Olympiad folks tend towards Engel's Problem Solving Strategies as it provides solutions.
• Amazon reviewer doing Olympiad problems recommends “harder” books like Schoenfeld, Zeitz, etc. here

YouTube Link! https://www.youtube.com/watch?v=spUNpyF58BY

• Code is here, https://github.com/3b1b/manim/blob/master/active_projects/fourier.py. Although it's basically his own library and he doesn't work at making it so understandeable or backwards compatible
• I don't really like the initial explanation though. Discussion on HN: https://news.ycombinator.com/item?id=16242103

To be improved

• Mathematician Timothy Gowers, The Importance of Mathematics hour long lecture, quite funny and approachable.
  • Why worry about coloring graphs? Seems kinda useless, a bit childish. Until you realize that it's exactly the same problem as timetabling, which comes up everywhere!
• Programmer/Statistician Evan Miller: The Mathematical Hacker and Don't Kill Math: Comments on Bret Victor's Scientific Agenda
  • Awesome and funny articles. Fortran is the “other parent” (compared to just coding) who has apparently been too busy at work to teach their child programmer math! Ouch…

Most engineers think that math is a rather boring subject, composed of rather trivial things like trigonometry and summing over infinitesimals (aka integrating). Also, most engineers were plagued with the fascination early on of making a robot move, or building their own bench out of wood, or creating mini flash games that could take the world by storm. When you've got something that's “hands on”, all you really need to do is plug n' chug the occasional formula or solve an inequality, right? There's even tools like Wolfram Alpha or your TI-89's that can solve all the ugly stuff you'd ever need to use! Why in the world would you ever need to really know math?

Well, as it turns out, there is a breed of person that got excited at an early age about things like proportions, puzzles, and the simple artistic beauty that math formulas sometimes hold, even graphically! In high school, they're known for tearing into their coursework and learning all they can about integrating and trig subbing, and often try and outdo their classmates in simplifying a problem the furthest! In college and beyond, they're known as mathematicians. They'll spend hours upon end, writing nary a word on their scratch piece of paper, but thinking deep thoughts about edge cases, identities, and elegant ways of making the massive problem they're dealing with a little bit simpler. Mathematicians make the world go round, and their advances in the last 100 years have largely been the reason that society is where it is today!

Turns out that mathematicians think about engineered robots, structures, and computers the same way! You just buy them and use them for their intended purpose, without any regard to the late nights debugging code and long ponderings about the optimum balance between form and function, and cost and completeness (what many an engineer stresses over)

Having a math/physics major as a roommate helped me a lot with learning these things. But honestly, this can be applied to just about any major vs another major in college! So, open your mind and learn from someone different than you!

• 16-811, CMU Robotics course with good notes on mathematical topics from an applied grad perspective

• DON'T KILL MATH! http://www.evanmiller.org/dont-kill-math.html. Excellent rebuttal! Need to understand the things he notes, like calculus of variations and stuff for optimization vs dumb knob tweaking in simulations.
  • Bret Victor heartily agrees: The ideal future representation for modeling systems won't resemble either analysis or simulation as we know it, but will descend from both.
• Really means, be smart about how you use and understand math. “[Math] symbols are a command line, we need a better interface!” from Bret Victor's Kill Math

This is exactly the problem: Saturday Morning Breakfast Cereal (SMBC) Physics comic:

• Excellent overview is Conrad Wolfram's TED Talk
• Also, excellent articles and examples of Computer Based Math from Wolfram blog posts and Conrad Wolfram
  • General gist is: “We don't want students to be third-rate computers; we want them to be first-rate problem solvers”
  • Also “We insist that the entire population learns how to do step 3 by hand. Perhaps 80% of doing math education at school is step 3 by hand and largely not doing steps 1, 2, and 4. And yet step 3 is the step that computers can do vastly better than any human at this point, so it's kind of bizarre that that's the way around we're doing things. Instead, I think we should be using computers to do step 3 and we should be using students to do steps 1, 2, and 4 to a much greater extent than we are.”

• What are the arguments against computer-based math?
• Is there a way to do fast theoretical problem solving? Input equations, input the variable you'd like to solve for…BAM, right?

• Highlights of Calculus (video series by Gilbert Strang), why it's important, etc.
• Strang's free calculus textbook: http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/

Linear Algebra with Applications is “better” than Introduction to Linear Algebra? Oh well…just time to learn linear algebra, here we go!

Linear algebra proof that matching two sequences using convolution (multiplying them and finding maximum) is the same as minimizing the mean-square error between the sequences.

• Gustavo mentioned solving a children's game where the sums of rows add up to certain values (kinda like sudoku but easier). That can be rewritten as a set of linear equations that you take the inverse of. BOOOOM SHAKA LAKA.

TO READ STILL

• Excellent visual presentation of what y = Ax is doing: http://blog.stata.com/2011/03/03/understanding-matrices-intuitively-part-1/
• Better Explained Linear Algebra. Got halfway through it…need to do 649! Comments.
• Excellent article on SVD applied to stochastic processes
• Spectral Clustering looks really cool, and seems to be related to kernel PCA (also really cool). All about that dimensionality reduction…
• Manga Guide to Linear Algebra
• Free Linear Algebra Textbook hopefully free is still good…
• Gilbert Strang's book is supposedly really good too. Video Lectures on OCW

• Notes on why they're cool and useful.
• http://math.stackexchange.com/questions/117021/what-do-the-eigenvectors-of-an-adjacency-matrix-tell-us.
  • Think Google and page linking. Seems like pretty crazy stuff, these eigens

• The $25 billion eigenvector (the story behind Google). 10 page paper.
  • Sort of interesting, but got lost pretty quickly in search engine theory. Pretty cool application though.
  • How the eigenvector actually works is still a mystery to me, but I'll hopefully figure it out soon.

• An optimized non-linear least squares solver that utilizes sparse datasets too. Not sure how to use it exactly, but it sounds really useful. Johnny Chung Lee writeup