Vector Calculus

Introduction Derivatives and coordinates Derivative of functions Inverse functions Coordinate systems Curves and Line Parametrised curves, lengths and arc length Line integrals of vector fields Gradients and Differentials Work and potential energy Integration in R2 and R3 Integrals over subsets of R2 Change of variables for an integral in R2 Generalization to R3 Further generalizations Surfaces and surface integrals Surfaces and Normal Parametrized surfaces and area Surface integral of vector fields Change of variables in R2 and R3 revisited Geometry of curves and surfaces Div, Grad, Curl and del Div, Grad, Curl and del Second-order derivatives Integral theorems Statement and examples Green's theorem (in the plane) Stokes' theorem Divergence/Gauss theorem Relating and proving integral theorems Some applications of integral theorems Integral expressions for div and curl Conservative fields and scalar products Conservation laws Orthogonal curvilinear coordinates Line, area and volume elements Grad, Div and Curl Gauss' Law and Poisson's equation Laws of gravitation Laws of electrostatics Poisson's Equation and Laplace's equation Laplace's and Poisson's equations Uniqueness theorems Laplace's equation and harmonic functions The mean value property The maximum (or minimum) principle Integral solutions of Poisson's equations Statement and informal derivation Point sources and delta-functions* Maxwell's equations Laws of electromagnetism Static charges and steady currents Electromagnetic waves Tensors and tensor fields Definition Tensor algebra Symmetric and antisymmetric tensors Tensors, multi-linear maps and the quotient rule Tensor calculus Tensors of rank 2 Decomposition of a second-rank tensor The inertia tensor Diagonalization of a symmetric second rank tensor Invariant and isotropic tensors Definitions and classification results Application to invariant integrals