So, basically, I am quite comfortable with mathematics in general. I enjoy tackling problems and thinking about them day-in and day-out when solving the mind boggling one's. Additionally, I also would like to learn the mathematics that is useful for CS, which I have studied undergraduate in and working towards an MSc too, specializing in AI with most modules which are more programming rigorous but less to minimum mathematical intuitiveness which kind of makes me feel unfulfilled and incomplete in learning those modules.
Now the thing is that the speed and magnitude at the which the information is being generated is humungous. And to keep with the coursework and the lectures and focusing on other things, I feel like I am not able to grasp the intuitive aspects of computer science subfields which are mathematically rigorous, either due to lack of time or the universities not being able to provide a separate module to study relevant mathematics (of course, due to lack of time). After my undergraduate in CS, I never got an opportunity to properly study mathematics which I am yearning to do since I started learning DL and it's applications to other domains.
Below is my undergraduate mathematics curriculum, now my issue is that how do I study the mathematics independently that is important to me for CS and also, for exploring Physics, one of the outliers in my interest cluster. I'd love to have some guidance on where to start considering my current knowledge in Maths.
Semester 1
1. Linear Algebra and Differential Equations
UNIT–I
Initial Value Problems and Applications
Exact differential equations - Reducible to exact.
Linear differential equations of higher order with constant coefficients: Non homogeneous
terms with RHS term of the type eax , sin ax, cos ax, polynomials in x, e ax V(x), xV(x)-
Operator form of the differential equation, finding particular integral using inverse operator, Wronskian of functions, method of variation of parameters.
Applications: Newton’s law of cooling, law of natural growth and decay, orthogonal
trajectories, Electrical circuits.
UNIT–II
Linear Systems of Equations
Types of real matrices and complex matrices, rank, echelon form, normal form, consistency
and solution of linear systems (homogeneous and Non-homogeneous) - Gauss elimination, Gauss Jordon, and LU decomposition methods- Applications: Finding current in the electrical
circuits.
UNIT–III
Eigen values, Eigen Vectors and Quadratic Forms
Eigen values, Eigen vectors and their properties, Cayley - Hamilton theorem (without proof),
Inverse and powers of a matrix using Cayley - Hamilton theorem, Diagonalization, Quadratic
forms, Reduction of Quadratic forms into their canonical form, rank and nature of the
Quadratic forms – Index and signature.
UNIT–IV
Partial Differentiation
Introduction of partial differentiation, homogeneous function, Euler’s theorem, total
derivative, Chain rule, Taylor’s and Mclaurin’s series expansion of functions of two
variables, functional dependence, Jacobian.
Applications: maxima and minima of functions of two variables without constraints and
Lagrange’s method (with constraints)
UNIT-V
First Order Partial Differential Equations
Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions, Lagranges method to solve the first order linear equations and the standard type methods to solve the non-linear equations.
Semester 2
2. Advanced Calculus
UNIT–I
Laplace Transforms: Laplace transforms of standard functions, Shifting theorems, derivatives and integrals, properties- Unit step function, Dirac’s delta function, Periodic function, Inverse Laplace transforms, Convolution theorem (without proof).
Applications: Solving ordinary differential equations (initial value problems) using Laplace
transforms.
UNIT-II
Beta and Gamma Functions: Beta and Gamma functions, properties, relation between Beta and Gamma functions, evaluation of integrals using Beta and Gamma functions.
Applications: Evaluation of integrals.
UNIT–III
Multiple Integrals: Double and triple integrals, Change of variables, Change of order of integration. Applications: Finding areas, volumes& Center of gravity (evaluation using Beta and Gamma functions).
UNIT–IV
Vector Differentiation: Scalar and vector point functions, Gradient, Divergence, Curl and their physical and geometrical interpretation, Laplacian operator, Vector identities.
UNIT–V
Vector Integration: Line Integral, Work done, Potential function, area, surface and volume integrals, Vector integral theorems: Greens, Stokes and Gauss divergence theorems (without proof) and related problems.
3. Statistical and Numerical Methods
UNIT–I
Random variables and Distributions:
Introduction, Random variables, Discrete random variable, Continuous random variable,
Distribution function, Expectation, Moment generating function, Moments and properties.
Discrete distributions: Binomial and geometric distributions. Continuous distribution: Normal distributions.
UNIT–II
Sampling Theory: Introduction, Population and samples, Sampling distribution of means (Sigma Known)-Central limit theorem, t-distribution, Sampling distribution of means (Sigma unknown)- Sampling distribution of variances – Chi-square and F- distributions, Point estimation, Maximum error of estimate, Interval estimation.
UNIT–III
Tests of Hypothesis: Introduction, Hypothesis, Null and Alternative Hypothesis, Type I and Type II errors, Level of significance, one tail and two-tail tests, Tests concerning one mean and proportion, two means-proportions and their differences-ANOVA for one-way classified data.
UNIT–IV
Algebraic and Transcendental Equations & Curve Fitting: Introduction, Bisection Method, Method of False position, Iteration methods: fixed point iteration and Newton Raphson methods. Solving linear system of equations by Gauss-Jacobi and Gauss-Seidal
Methods. Curve Fitting: Fitting a linear, second degree, exponential, power curve by method of least squares.
UNIT–V
Numerical Integration and solution of Ordinary Differential equations: Trapezoidal rule- Simpson’s 1/3 rd and 3/8 th rule- Solution of ordinary differential equations by Taylor’s series,Picard’s method of successive approximations, Euler’s method, Runge-Kutta method (second and fourth order)
Semester 3
4. Complex Variables and Fourier Analysis
UNIT–I
Functions of a complex variable: Introduction, Continuity, Differentiability, Analyticity, properties, Cauchy, Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions-Milne-Thompson method
UNIT-II
Complex integration: Line integral, Cauchy’s integral theorem, Cauchy’s integral formula, and Generalized Cauchy’s integral formula, Power series: Taylor’s series- Laurent series, Singular points, isolated singular points, pole of order m – essential singularity, Residue, Cauchy Residue theorem (Without proof).
UNIT–III
Evaluation of Integrals: Types of real integrals:
Bilinear transformation- fixed point- cross ratio- properties- invariance of circles.
UNIT–IV
Fourier series and Transforms: Introduction, Periodic functions, Fourier series of periodic function, Dirichlet’s conditions, Even and odd functions, Change of interval, Half range sine and cosine series. Fourier integral theorem (without proof), Fourier sine and cosine integrals, sine and cosine, transforms, properties, inverse transforms, Finite Fourier transforms.
UNIT–V
Applications of PDE: Classification of second order partial differential equations, method of separation of variables, Solution of one-dimensional wave and heat equations.
Personally, I would say Numerical Analysis is where anyone comfortable with computers will shine. Ordinary Differential Equations, Stochastic Differential Equations (stochastic methods in general), and Markov Chains can all be deeply explored with numerical analysis. These have applications in physics, finance, engineering, and medicine, so it really comes down to finding a problem that you find interesting.
With your math background and programming experience, it wouldn't take long for you to make decent progress exploring this field.
This first part answers your question about learning math for fun (which of course is biased because this is where I have the most fun combining math and programming).
If you're mostly wanting to focus on math that will most directly make you a better programmer, then you'll probably want to start by looking at graph theory and abstract algebra.