Kuttler K. Lecture for Partial Differential Equations

2003. - 151 pages.
Review Of Advanced Calculus.
Continuous Functions Of One Variable.
Exercises.
Theorems About Continuous Functions.
The Integral.
Upper And Lower Sums.
Exercises.
Functions Of Riemann Integrable Functions.
Properties Of The Integral.
Fundamental Theorem Of Calculus.
Exercises.
Multivariable Calculus.
Continuous Functions.
Sufficient Conditions For Continuity.
Exercises.
Limits Of A Function.
Exercises.
The Limit Of A Sequence.
Sequences And Completeness.
Continuity And The Limit Of A Sequence.
Properties Of Continuous Functions.
Exercises.
Proofs Of Theorems.
The Concept Of A Norm.
The Operator Norm.
The Frechet Derivative.
Higher Order Derivatives.
mplicit Function Theorem.
The Method Of Lagrange Multipliers.
Taylor's Formula.
Weierstrass Approximation Theorem.
Ascoli Arzela Theorem.
Systems Of Ordinary Differential Equations.
The Banach Contraction Mapping Theorem.
C1 Surfaces And The Initial Value Problem.
First Order PDE.
Quasilinear First Order PDE.
Conservation Laws And Shocks.
Nonlinear First Order PDE.
Wave Propagation.
Complete Integrals.
The Laplace And Poisson Equation.
The Divergence Theorem.
Balls.
Polar Coordinates.
Poisson's Problem.
Poisson's Problem For A Ball.
Does It Work In Case f = 0?
The Case Where f 0, Poisson's Equation.
The Half Plane.
Properties Of Harmonic Functions.
Laplace's Equation For General Sets.
Properties Of Subharmonic Functions.
Poisson's Problem Again.
Maximum Principles.
Elliptic Equations.
Maximum Principles For Elliptic Problems.
Weak Maximum Principle.
Strong Maximum Principle.
Maximum Principles For Parabolic Problems.
The Weak Parabolic Maximum Principle.
The Strong Parabolic Maximum Principle.