Author(s): B. Allanach, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part IA Lecture Notes
Publisher: University of Cambridge
Year: 2015
Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; calculus; analysis; differential; integral; single variable; multiple variables; multivariable; vector; tensor; one variable; many variables
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
