Vector Calculus

Abstract

Introduction 1. Differential geometry of curves 1.1. Parametrised curves and arc-length 1.2. Curvature and torsion 1.3. Radius of curvature 1.4. Gauss, pizza and curvature of surfaces (non-examinable) 2. Coordinates, Differentials and Gradients 2.1. Differentials and first order changes 2.2. Coordinates and line elements 2.3. The gradient operator 2.4. Computing the gradient 2.5. Summary 3. Integration over lines, surfaces and volumes 3.1. Line integrals 3.2. Conservative forces and exact differentials 3.3. Integration in R2 3.4. Integration over surfaces 4. Divergence, Curl and the Laplacian 4.1. Definitions 4.2. Topology via calculus (non-examinable) 5. Integral theorems 5.1. Green’s theorem: statement and examples 5.2. Stokes’ theorem: statement and examples 5.3. Möbius strips and Stokes (non-examinable) 5.4. Divergence theorem: statement and examples 5.5. Noether’s theorem (non-examinable) 5.6. Sketch proofs 6. Maxwell’s Equations 6.1. Brief introduction to electromagnetism 6.2. Integral formulation 6.3. Electromagnetic waves 6.4. Electrostatics and Magnetostatics 6.5. Gauge invariance (non-examinable) 7. Poisson’s equation