Reddit reviews Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.

For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,

By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.

Gilbert Strang wrote one of the standard textbook in linear algebra and teaches out of it in his class on MIT OpenCourseware.

I preferred Shifrin's Multivariable Mathematics and there also videos of him teaching the class. But the books have different sensibilities and I thought one worked well as a back up and different perspective to the other.

Plus, in Shifrin's text, multivariable calculus and linear algebra are treated at the same time, which made a lot of sense at the time. It makes a lot about the two subjects make more sense.

• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
  
  
• Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
  
  
• Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds
  
  
• Advanced Calculus of Several Variables (Dover Books on Mathematics)
  
  
• Calculus of Several Variables (Undergraduate Texts in Mathematics)

Linear Algebra (preferably proof based, theoretical) Introduction to Proofs (usually a perquisite for Linear Algebra) and Multivariable Calculus

Here’s a pdf of the textbook: http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

Note that this pdf does not cover differential forms but continue to read if you want to know more about the topic!

Now, if you want to learn more about differential forms, read chapter 8 of this textbook: https://www.amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/dp/047152638X

A more economical way to learning differential form is to watch MATH 3510 videos on YouTube, particularly on differential forms. MATH 3500-3510 is a rigorous year sequence that covers many topics including differential geometry. These lectures go by the book I listed above. The prerequisite for this course is Calculus II but as a caveat, this course is not easy to self-study.

Haha, studying for the GRE, I know that now, but I was never aware of how important it was. The most topology I dealt with was in my complex analysis class and in my multivariable class where we used this book. That class initiated my masochistic addiction to math.

I say masochistic because I also studied biology to some depth. I was always rushing to catch up with one major or the other. So point-set topology probably got lost in crossfire of a laundry list of other classes I had to take. But I don't regret bio: I want to do applied math focusing on biological problems, i.e. dynamical systems, high dimensional networks, and other problems motivated by bioinformatics, computational biology, and biophysics. Ideally, I can get into Duke or UCLA's biomath programs; they seem pretty well established from the research I've done. However, they're definitely "reach" schools. Not putting all of my eggs in those baskets.