The Ideal First Year Undergraduate Curriculum for a Mathematics Autodidact

Question

I know that the following question may not be appropriate for this forum but please do hear me out.

I apologize in advance for the length of this post.

Due to the unfolding of certain events in my life i ended up doing a bachelors in Electrical Engineering but my interest always lied in Mathematics however due to certain complications i could not switch my Major and ended up being quite miserable. I still wish to learn and not waste the small window i have left to have a respectable career in Mathematics.

Therefore I would highly appreciate it if the many users of MSE could suggest a FIRST YEAR UNDERGRADUATE MATHEMATICS CURRICULUM for someone like me, to make it easier for you to answer i have outlined (briefly) below my current background and the things that i have tried to learn on my own.

• I have covered the typical Mathematics Course for engineers that involve Single and Multi-variable Calculus,Linear Algebra,Differential Equations,Complex Variables and Transforms etc.

NOTE PLEASE THAT IN ALL OF THE ABOVE COURSES THERE WAS NO EMPHASIS ON RIGOROUS PROOFS WHAT SO EVER.

• I have largely covered How to Prove it by Daniel J Velleman and have gotten comfortable with the idea of a proof and how to construct elementary proofs.
• I am currently working my way through Linear Algera Done Right by Sheldon Axler albeit with some difficulty in the later Chapters.

furthermore i know that there are very good source out there such as MIT-OCW, Quant Academy etc. that would could serve as adequate guides, but having seen how useful MSE has been for me in these $8+$ months I am hopeful for an answer more suited to my needs.

Answer 1

Linear Algebra:

1. Intro to sets, Vector Spaces: Bases and Dimensions, Steinitz Theorems, matrices.
2. Classification of finite dimensional Endomorphisms[3-6] (diagonalization, Jordan normal form, Frobenius form etc). You will need to know cyclic spaces, nilpotent endomorphisms etc. Intro to rings for minimal polynomials.
3. Bilinear forms, quadratic forms, etc.
4. Orthogonality, general scalar products,
5. Isometries, projections, self-adjoint endomorphisms. Orthogonal polynomials.
6. "Famous" matrix groups, $\text{GL}_n$, $\text{O}_n$, etc. Permutations?

Analysis:

1. Field axioms, Real numbers, Sequences, Limits, Series, Continuity, Scalar function differentiation/integration.
2. Normed spaces, Metric spaces, Sequences in metric spaces, Completeness of metric spaces.
3. Multivariate calculus: Fréchet derivative, Jacobi matrices, Hessian, etc.
4. Implicit Function Theorem and Local Inverse Theorem.
5. Vector fields, step functions, multivariate integration*.
6. The Riemann integral, Fubini's Theorem.
7. Intro calculus of variation. Curve integrals etc.

Answer 2

You have covered much of useful undergraduate mathematics content according to you. Great start. My one piece of advice is "do what you love". Please don't compare yourself to "graduate level Mathematics Student" because your study of mathematics should not be a competitive race. Read a general overview of mathematics such as Princeton Companion to Mathematics and decide what areas of mathematics you want to study. Then practice, practice, practice. When you encounter difficult subjects, it is okay to skip them at first while going on to other subjects. You can always go back later and try again.

Answer 3

What do you want to do when you work later on?

1. Write proofs & develop theory (pure math)
2. build stuff / program computers (engineering)
3. measure things/model the world (sciences)

What courses you should take will vary depending on that answer.