electiveHQ

This module introduces the fundamental concepts and methods in the differentiation and integration of vector-valued functions and also provides an introduction to the Calculus of Variations. [Fundamentals] Vectors and scalars, the dot and cross products, the geometry of lines and planes, [Curves in three-dimensional space] Diferentiation of curves, the tangent vector, the Frenet- Serret formulas, key examples of Frenet-Serret systems to include two-dimensional curves, and the helix, [Taylor's theorem in one and several variables] Conditions for the convergence of Taylor series, practical computations, extension to Taylor's theorem in several variables, the connection with the differential of a multi-variable function, [Partial derivatives and vector fields] Introduction to partial derivatives, scalar and (Cartesian) vector fields, the operators div, grad, and curl in the Cartesian framework, applications of vector differentiation in electromagnetism and fluid mechanics, [Mutli-variate integration] Area and volume as integrals, integrals of vector and scalar fields, Stokes's and Gauss's theorems (statement and proof), [Consequences of Stokes's and Gauss's theorems] Green's theorems, the connection between vector fields that are derivable from a potential and irrotational vector fields, [Curvilinear coordinate systems] Basic concepts, the metric tensor, scale factors, div, grad, and curl in a general orthogonal curvilinear system, special curvilinear systems including spherical and cylindrical polar coordinates Further topics may include: Introduction to differential forms, exact and inexact differential forms, [Advanced integration] Integrating the Gaussian function using polar coordinates, the gamma func- tion, the volume of a four-ball by appropriate coordinate parameterization, the volume of a ball in an arbitrary (finite) number of dimensions using the gamma function, [Applications in general relativity] Lengths and volumes in curved spacetime. [Fluid mechanical application] Incompressible flow over a wavy boundary, [Calculus of variations] Constrained variations.