Your answers and comments would be really appreciated as they will help me for self-studying. I graduated form the college with BSc in chemical engineering. I had the following maths classes:
Zeroth (Preparatory) Year:
- First Semester: Algebra.
- Second Semester: Trigonometry.
First Year:
- First Semester: Calculus I. (Limits with one variable, basic differentiation)
- Second Semester: Calculus II. (only proper single integrals that can be solved with $u$-substitution, trigonometric substitution, by parts, or some other basic techniques - nothing was hard).
Second Year:
- First Semester: No maths class as per the study plan.
- Second Semester: Probability and Statistics. (Only basic things such as addition rule $P(A \text{ or }B),$ multiplication rule $P(A \text{ and }B)$, complimentary rule $P(A^C)=1-P(A)$, conditional rule $P(B|A)=P(A \text{ and }B)/P(A)$. Then simple things like correlation and linear regression for $y=ax+b$ and nothing about other kinds of regression like multiple linear regression or logistic regression).
Third Year:
- First Semester: Two courses (as per the study plan):
Calculus III (Dot product & cross product of vectors, then testing (without evaluating) the convergence and divergence with root test and ratio test (no other test), then double integrals and triple integrals).
Differential Equations (variable separable, exact, higher order homogenous and non-homogenous DE, Laplace transform, Fourier series)
- Second Semester: Engineering Mathematics (curl & div & grad, line integrals, Green's theorem, Stock's theorem, Contour integrals)
Fourth Year: (No maths courses in both semesters).
First, the study plan, I believe, was not good. Some semesters we study things for elementary schools. And some semesters, we study very hard things where we do not have the basics of those things. Also, the first semester in the third year was heavy as I was focusing in chemical engineering courses, but with 2 maths courses, literally PAINFUL!
Second, the instructors were not so good. Even when a student ask a question, they struggle and say: "mmmm, let me check and inform you later", then nothing.
I am not that bad at maths, and I am very interested to review what I know already and study somethings new to me. The idea I am having and aiming to achieve is: Mentioning the mathematical area, and listing (under that area) the "names/titles" of the basic axioms and formulae that each graduate student MUST know.
See the following (as illustration) of what I am aiming:
- Algebra:
-- the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom.
-- quadratic formula for polynomials.
-- binomial theorem.
-- rational root theorem.
-- Partial fraction decomposition
-- AND MUCH MORE
- Trigonometry:
-- definition of trigonometric function on the unit circle.
-- sine and cosine rules.
-- half and double angle formulae.
-- sum to product and product to sum formulae.
-- AND MUCH MORE
- Linear Algebra:
-- determinants.
-- Rank-Nullity theorem.
-- Cayley-Hamilton theorem.
-- AND MUCH MORE
- Number Theory:
-- Fermat's Little Theorem.
-- Euler's totient theorem.
-- Chinese Remainder theorem.
-- AND MUCH MORE
- Group Theory:
-- Group axioms.
-- AND MUCH MORE
- Combinatorics:
-- inclusion–exclusion principle.
-- AND MUCH MORE
- Calculus:
-- Epsilon-Delta definition of limits.
-- First and second derivative test
-- L'Hospital's Rule.
-- Fundamental theorem of calculus
-- Squeeze theorem.
-- Chain Rule.
-- Cauchy's theorem
-- Gauss' Divergence theorem
-- AND MUCH MORE
I understand that maths is a wide science, and nobody can list everything. However, once one person mentions and list some of the basic things, others may add in the comment section. Then I can go and search by myself.
I am hoping from you to initiate by the area of your interest. There are more areas that I did not mention above, such as topology, graph theory, logic, etc. but I wish you mention if that is the area you like.
Again, I am asking for the things that everyone interested in that area must know.
This post, hopefully, will be one of the references especially for those who want to study.
THANKS A LOT!
Still I did not get any suggestion. Can you suggest?