Guide to Mathematical Methods

Part 1 Functions: sets and intervals; functions; limits and continuity; polynomials; rational functions; trigonometric functions; exponential and logarithmic functions; hyperbolic functions; composite and inverse functions. Part 2 Differentiation: introduction; derivatives of combinations of functions; derivatives of trigonometric functions; derivatives of exponential and logarithmic functions; higher derivatives; power series expansion. Part 3 Integration: area and definite integrals; speed and distance, force and work; the fundamental theorem of calculus; standard integrals; integration of substitution; integration by parts; partial fractions; systematic integration of rational functions; rational trigonometric functions; improper integrals. Part 4 Linear equations, determinants and matrices: introduction; determinants; three or more simultaneous equations; matrices; square matrices; numerical solution of linear equations. Part 5 Vectors: introduction; coordinate systems; the algebra of vectors; unit vectors and direction cosines; scalar products; vector products; triple products; lines and planes; vector equation of curves in space; differentiation of vector functions; arclength. Part 6 Functions of two variables: introduction; the standard family of functions; graphical representation; functions of three or more variables; partial derivatives; chain rules; directional derivatives; higher partial derivatives; maxima and minima. Part 7 Line integrals and double integrals: vector fields; line integrals; properties of line integrals; conservative fields; double integrals; change of variables; Green's theorem. Part 8 Complex numbers: introduction; the algebra of complex numbers; solution of equations; equalities and inequalities; polar form of complex numbers; exponential form of complex numbers; trigonometric identities. Part 9 Differential equations: introduction; differential equations of type 1 (separable); differential equations of type 2; differential equations of type 3 (exact); differential equations of type 4 (linear); an alternative way of solving first order linear equations; solution of second order differential equations; complementary function of second order differential equations; general solution of non-homogeneous equation; initial and boundary conditions.