Handbook

Many phenomena and problems, both real-world and theoretical, give rise to relationships between multiple independent or dependent variables, and possibly both. This topic extends key concepts from single-variable calculus to multivariable cases, providing a mathematical language to describe these relationships. This begins with a review of the geometry of space, including vectors, lines and planes, introducing the idea of limits and derivatives to this setting. Functions of multiple variables are introduced and described using partial derivatives, giving rise to the gradient, directional derivatives and multivariable versions of the chain rule, Taylor series, and optimisation techniques, including Lagrange multipliers. The topic also introduces multivariable integrals, including the use of polar, cylindrical and spherical co-ordinate systems to evaluate these. The ideas considered are drawn together in the study of vector-valued functions, introducing the divergence, curl, Laplacian, line integrals and surface integrals. This culminates in multivariable extensions of the fundamental theorem of calculus, including the gradient, Green's, Stokes', and divergence theorems.