My undergrad used: Calculus: Early Transcendental for single variable, multi variable, and some of PDEs. I believe that it gives a pretty good overview of the important theorems. You can also pair this with Paul's Notes. I would then look into linear algebra, my courses used Linear Algebra and its Applications. Next up would probably be statistics or probability (A First Course in Probability).
I will say: MFE programs don't usually cover basic calculus - that is usually assumed knowledge. This material would be used to prepare you for a MFE program and should not be considered a substitute. MFE programs should cover other math (stochastic calc, applications of probability, PDEs, matrix analysis, etc). If you are coming from a STEM-less background, I would self study the basics and then apply to a legitimate MFE program. This will show employers that you can actually perform math at a high level. I would be pretty skeptical of an applicant who has claimed to have taught themselves 5-6 years of math with no proof from a third party. Even if you could accomplish this, I would bet the employer would rather go with the applicant who could provide more formal proof of education (a diploma, transcript, and gpa).
I think most MFEs expect you to have a basic foundation in Math, Stats, and Programming. Finance knowledge is always considered a good-to-have, but not a must-have like the other three pillars of quantitative finance.
I believe that the purpose of MFE is to build on that foundation:
- first, you go from basic concepts that you know to advanced concepts,
- then you apply those concepts in Finance
- then you try to figure out the next step..i.e. how can you go even beyond the teachings of the class (for your career, research, etc..)
Therefore, starting with entry-level calculus doesn’t seem like the right approach for a program, to me. MFE programs must focus on making sure the students have up-to-date knowledge about the field they wish to work in before leaving the school so that they can hit the ground running in the industry. That’s why industry projects/internships are a part of the curriculum.
If any MFE program starts with teaching any of the pre-requisites in let’s say the first semester, etc.. then that’s a big red flag for me
You should cover the basics through the pre-MFE courses which is the right place to start as mentioned above by professor @Daniel Duffy.
a book that i used for revision before starting my mfe program was "Vector Calculus, Linear Algebra, and Differential Forms" by hubbard and hubbard
i've not seen this book recommended enough but it covers a lot of things in a very accessible manner and the thing i liked about it the most is that it uses linear algebra as a setting for explaining topics such as vector calculus
there's a small but really well-done section called "Probability and the singular value decomposition" — now if you've done an mfe degree, you'll know how important those concepts are
nb: this is probably not a very good first book if you've never seen these topics before; i have an undergad in engineering and was looking for something that would give me a holistic refresher and intuition on a lot of topics rather than start with the very basics like spivak's calculus or strang's linear algebra would've done
Buy An Introduction To Linear Analysis by Kreider Donald L, Kuller Robert G, Ostberg Donald R, & Perkins Fred W (ISBN: ) from Amazon's Book Store. Everyday low prices and free delivery on eligible orders.
This is perhaps the single best advanced calculus text on the market and has been for the last fifteen or twenty years. It used to be the prescribed text (postscript: at Cornell) for freshmen who had earned a 5 in AP calculus (BC) in high school. It's also used at other universities by freshmen with similar backgrounds. But I'm not sure it's the best text for quant wannabes who may not have stellar math backgrounds.