Lecture notes on the Runge-Kutta method of numerical integration, Taylor series expansion, formal derivation of the second-order method, Taylor expansion using indicial notation and summation convention, Taylor expansion of a vector function of a vector, Nyström’s third order method, series expansion using indicial notation, series expansion using vector notation, condition equations, Nyström’s third order algorithm, Nyström’s fourth order algorithm, Nyström’s fifth order algorithm, and R-K-N sixth order algorithm.