title: Green's Functions with Applications
author:  Dean G. Duffy 
description: Roughly speaking, Green's functions constitute infinitesimal matrix coefficients that one can use to solve linear nonhomogeneous differential equations in an approach alternative to that which depends on eigenvalue analysis. These techniques receive a mention in many books on differential equations. Duffy goes much further toward exposing the detailed workings of important examples (wave equation, heat equation, Hemholtz equation on various domains)Many plots help the reader picture the behavior of these functionsA valuable sourcebook. <br />- CHOICE Magazine, March 2002
zlib_text: 2C2 Multivariate Calculus
  
  Michael D. Alder
  
  November 13, 2002
  
  2
  
  Contents
  
  1
  
  Introduction
  
  5
  
  2
  
  Optimisation
  
  7
  
  2.1
  
  The Second Derivative Test
  
  . . . . . . . . . . . . . . . . . . .
  
  7
  
  3
  
  Constrained Optimisation
  
  15
  
  3.1
  
  Lagrangian Multipliers . . . . . . . . . . . . . . . . . . . . . .
  
  15
  
  4
  
  Fields and Forms
  
  23
  
  4.1
  
  Definitions Galore . . . . . . . . . . . . . . . . . . . . . . . . .
  
  23
  
  4.2
  
  Integrating 1-forms (vector fields) over curves. . . . . . . . . .
  
  30
  
  4.3
  
  Independence of Parametrisation
  
  . . . . . . . . . . . . . . . .
  
  34
  
  4.4
  
  Conservative Fields/Exact Forms . . . . . . . . . . . . . . . .
  
  37
  
  4.5
  
  Closed Loops and Conservatism . . . . . . . . . . . . . . . . .
  
  40
  
  5
  
  Green’s Theorem
  
  47
  
  5.1
  
  Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  
  47
  
  5.1.1
  
  Functions as transformations . . . . . . . . . . . . . . .
  
  47
  
  5.1.2
  
  Change of Variables in Integration
  
  . . . . . . . . . . .
  
  50
  
  5.1.3
  
  Spin Fields
  
  . . . . . . . . . . . . . . . 