FULL LEARNING PATH

Calculus and Advanced Mathematics

01

Mathematical Foundations

Number systems, algebra, equations, inequalities, coordinate geometry, mathematical notation and essential prerequisite ideas.

Explore → 02

Functions and Elementary Models

Function notation, domain and range, graphs, transformations, polynomial, rational, exponential, logarithmic and trigonometric functions.

Explore → 03

Change and Limits

The meaning of a limit, one-sided limits, algebraic methods, infinite limits and limits at infinity.

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Continuity

Continuity at points and on intervals, types of discontinuity, and the relationship between limits and function values.

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Differentiation

Derivatives from first principles, tangent slopes, \(dy/dx\), differentiation rules, implicit differentiation and applications.

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Integration

Accumulation, antiderivatives, definite and improper integrals, the Fundamental Theorem and integration techniques.

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Sequences and Series

Sequences, convergence, infinite series, convergence tests, power series and Taylor and Maclaurin expansions.

Explore → 08

Ordinary Differential Equations

First- and higher-order ODEs, initial-value problems, systems, phase portraits, equilibria and stability.

Explore → 09

Multivariable Calculus

Functions of several variables, partial and directional derivatives, gradients, Jacobians, Hessians and multiple integrals.

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Vector Calculus

Vector fields, line and surface integrals, divergence, curl, conservative fields and the major integral theorems.

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Linear Algebra for Calculus and Models

Vectors, matrices, linear systems, transformations, determinants, eigenvalues, eigenvectors and matrix stability.

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Mathematical Modelling and Dynamical Systems

Building models from rates and mechanisms, growth and decay, population and epidemic models, equilibria, stability and bifurcation ideas.

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Probability and Random Variables

Probability spaces, conditional probability, random variables, distributions, expectation, variance and joint distributions.

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Stochastic Processes

Discrete- and continuous-time Markov chains, transition probabilities and rates, generators, Poisson processes and Brownian motion.

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Stochastic Differential Equations

Drift and diffusion, formulation and interpretation of SDE models, solution behaviour and Euler–Maruyama approximation.

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Stochastic Calculus

Quadratic variation, Itô integrals, Itô's lemma, stochastic chain rules and analytical tools for diffusion processes.

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Numerical Analysis and Scientific Computing

Root finding, interpolation, numerical differentiation and integration, ODE solvers, errors, stability and stochastic simulation.

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Real Analysis

Completeness, rigorous limits and continuity, sequences and series, differentiability, Riemann integration and convergence of functions.

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Complex Analysis

Complex numbers and functions, complex differentiation, analytic functions, contour integration, Cauchy's theorem, residues and conformal maps.

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Partial Differential Equations

First- and second-order PDEs, heat, wave and Laplace equations, boundary conditions, separation of variables and Fourier methods.

Explore → 21

Measure Theory and Advanced Integration

σ-algebras, measures, measurable functions, Lebesgue integration, convergence theorems and the rigorous foundation of modern probability.

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One continuous mathematical story

\[\text{Foundations}\to\text{Functions}\to\text{Limits}\to\text{Continuity}\to\text{Differentiation}\to\text{Integration}\to\text{Series}\]\[\downarrow\]\[\text{ODEs}\to\text{Multivariable}\to\text{Vector Calculus}\to\text{Linear Algebra}\to\text{Modelling}\]\[\downarrow\]\[\text{Probability}\to\text{Stochastic Processes}\to\text{SDEs}\to\text{Stochastic Calculus}\]\[\downarrow\]\[\text{Numerical Analysis}\to\text{Real Analysis}\to\text{Complex Analysis}\to\text{PDEs}\to\text{Measure Theory}\]

The path begins intuitively and progressively develops the computational, applied and rigorous mathematics needed for advanced analysis and stochastic modelling.