Miashs Bachelor's Mathematics Course

Lesson 1 • Logic and Propositional Calculus

Introduces truth values, connectives, and quantifiers as the grammar of mathematics. Mastery here underpins every proof technique in the chapter.

Lesson 2 • Integers and Divisibility

Develops divisibility, prime factorization, and modular arithmetic as concrete proof practice. Connects abstract logic to number-theoretic results.

Lesson 3 • Proof Techniques

Trains direct proof, contrapositive, contradiction, and induction as core argumentative tools. Students apply each method to number-theoretic statements.

Lesson 1 • Series and Sequences

Analyzes convergence of numerical sequences and infinite series. Introduces power series and Taylor expansions as analytic approximation tools.

Lesson 2 • Limits and Continuity

Defines the limit rigorously and characterizes continuous functions on intervals. Establishes the analytic foundation for derivatives and integrals.

Lesson 3 • Integral Calculus

Constructs the Riemann integral and proves the Fundamental Theorem of Calculus. Develops substitution and integration by parts as primary techniques.

Lesson 1 • Vectors and Matrix Operations

Introduces vectors in n-dimensional space and matrix arithmetic including multiplication and inversion. Provides computational tools used throughout the chapter.

Lesson 2 • Inner Product Spaces

Introduces dot products, norms, and orthogonality in abstract inner product spaces. Gram-Schmidt orthogonalization and least-squares applications are developed.

Lesson 3 • Systems of Linear Equations

Applies Gaussian elimination and row reduction to solve and classify linear systems. Links solution structure to the rank of the coefficient matrix.

Lesson 4 • Linear Maps and Eigentheory

Studies linear transformations, their matrix representations, and spectral decomposition. Diagonalization is applied to differential equations and data analysis.

Lesson 1 • Topology of Euclidean Space

Defines open sets, closed sets, and limits in R^n to ground multivariable analysis. Provides the topological language needed for continuity and differentiability.

Lesson 2 • Vector Calculus Theorems

Unifies line integrals, surface integrals, and the classical theorems of Green, Stokes, and Gauss. Students interpret these theorems physically and apply them to flux computations.

Lesson 3 • Multiple Integrals

Constructs double and triple integrals via iterated integration and change of variables. Fubini's theorem and polar, cylindrical, and spherical coordinates are applied.

Lesson 4 • Partial Derivatives and Differentiability

Develops partial derivatives, the gradient, and the total derivative as the multivariable analog of the single-variable derivative. Applies the chain rule to composite maps.

Lesson 1 • Random Variables and Distributions

Introduces discrete and continuous random variables, their distributions, and key parameters. Covers the most important named distributions and their properties.

Lesson 2 • Probability Spaces and Axioms

Defines sample spaces, events, and Kolmogorov axioms as the formal basis of probability. Conditional probability and independence are introduced with combinatorial examples.

Lesson 3 • Joint Distributions and Independence

Analyzes joint, marginal, and conditional distributions for multiple random variables. Covariance and correlation quantify linear dependence between variables.

Lesson 4 • Statistical Inference

Develops point estimation, confidence intervals, and hypothesis testing from a frequentist perspective. Connects theoretical distributions to real data analysis.

Lesson 1 • Qualitative and Numerical Methods

Analyzes equilibria, stability, and long-term behavior without explicit solutions. Introduces Euler and Runge-Kutta methods for numerical approximation.

Lesson 2 • Laplace Transform Methods

Uses the Laplace transform to convert ODEs into algebraic equations for systematic solution. Handles discontinuous forcing functions and impulse inputs.

Lesson 3 • First-Order Differential Equations

Classifies and solves separable, linear, and exact first-order ODEs. Introduces direction fields and existence-uniqueness theory for initial value problems.

Lesson 4 • Second-Order Linear ODEs

Solves homogeneous and non-homogeneous second-order linear equations with constant coefficients. Applies results to mechanical and electrical oscillation models.

Lesson 1 • The Real Number System

Constructs the reals via completeness and establishes supremum, infimum, and the Archimedean property. These properties underpin every convergence argument in the chapter.

Lesson 2 • Continuity and Uniform Continuity

Distinguishes pointwise and uniform continuity and proves key theorems on compact intervals. Connects topological compactness to analytic properties of functions.

Lesson 3 • Sequences and Series Rigorously

Reproves sequence and series convergence with full epsilon-N rigor. Cauchy sequences and completeness are central tools.

Lesson 4 • Differentiation Theory

Reproves the Mean Value Theorem and its consequences with full rigor. L'Hopital's rule and Taylor's theorem with remainder are derived analytically.

Lesson 1 • Numerical Integration

Derives quadrature rules from interpolation and analyzes their accuracy via error formulas. Adaptive and Gaussian quadrature extend precision for smooth integrands.

Lesson 2 • Floating-Point Arithmetic and Error

Explains IEEE floating-point representation, rounding, and error propagation in computations. Establishes the vocabulary of absolute, relative, and round-off error.

Lesson 3 • Numerical Linear Algebra

Applies direct and iterative solvers to large linear systems and analyzes their stability. QR decomposition and singular value decomposition are introduced for data applications.

Lesson 4 • Interpolation and Approximation

Constructs polynomial interpolants and splines and bounds interpolation error. Least-squares polynomial fitting is derived from linear algebra.