Mathematics

Differential equations and series

First-order linear, harmonic oscillator, Taylor and geometric series.

Basics

Differential equations

They tie rates to the state. Separable, first-order linear (integrating factor), constant-coefficient second order (characteristic equation). Initial values pick the solution. Motion, circuits, and growth land here.

Series

Taylor is a local polynomial. A geometric series sums closed for |r|<1. Do not use it past the radius of convergence. Fourier unpacks periodic functions into sines.

Formulas

Exponential growth/decay

y′ = k y ⇒ y = y₀ e^{k t}

k>0 grows, k<0 decays. Half-life is ln 2 / |k|.

Symbols

k rate constant

Simple harmonic motion

x″ + ω² x = 0, ω = √(k/m)

Spring, or a small pendulum (approx.). Period T=2π/ω.

Symbols

ω angular frequency

Infinite geometric series

∑_{n=0}^{∞} r^n = 1/(1−r) (|r|<1)

Do not use this sum when |r|≥1.

Symbols

r common ratio

In this field

Algebra and geometry

Equations, exponentials and logs, Pythagoras, circles.

Trigonometry and calculus

Trig ratios, identities, derivatives, integrals.

Statistics and linear algebra

Mean, variance, Bayes, matrices, eigenvalues.

Discrete, complex, series

Sequences, series, complex numbers, combinatorics.

Limits and number-theory sketch

Limits, continuity, primes, congruences.

Vectors and space

Vectors, dot and cross products, planes, and distance in 3D.

Conic sections

Parabola, ellipse, hyperbola, eccentricity, foci, and an orbit sketch.

Fourier sketch

Sines that rebuild a periodic function, coefficients, spectra, and a discrete-transform note.

Probability, continued

Expected value, variance, normal density, and a CLT one-liner.

Graph-theory sketch

Vertices and edges, degree, paths, Euler, and the adjacency matrix.