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