11Applications of the Divergence Theorem
MATH 22C
1. Derivation of the Rankine-Hugoniot Jump Conditions
In this section we apply the Divergence Theorem to derive the RankineHugoniot (RH) jump conditions, which we already used to derive the
shock curves of
7The Nonlinear Wave Equation
and the
Interaction of Waves
MATH 22C
1. Introduction
Just as the nonlinear advection equation looks the same
as the linear advection equation ut + cux = 0 except that
the speed of sound c depends on the solution u, so also
th
8Simple Waves
and the
Nonlinear Theory of Sound
MATH 22C
1. Introduction
In the last section we showed that the nonlinear wave equation admits nonlinear elementary waves that propagate to
the left and to the right. But, because the equations are
nonlinear
9Shock Waves
and the
Riemann Problem
MATH 22C
1. Introduction
The purpose of this section is to solve the so called Riemann problem for Burgers equation and for the p-system.
The Riemann problem is the initial value problem when
the initial data consists
10Maxwells Equations
Stokes Theorem
and the
Speed of Light
MATH 22C
1. Introduction
In the next four sections we present applications of Stokes
Theorem and the Divergence Theorem. In the rst application, presented here, we use them to give a physical inte
6Energy Methods And The Energy of Waves
MATH 22C
1. Conservation of Energy
We discuss the principle of conservation of energy for ODEs,
derive the energy associated with the harmonic oscillator,
and then use this guess the form of the continuum version
of
5-LINEARIZING EQUATIONS ABOUT REST POINTS:
The Speed of Sound, the Speed of Light
and
The Fundamental Dierence Between Them
MATH 22C
1. Points of equilibrium=rest points
An equilibrium point or rest point of an equation is a constant state solution of th
1-Introduction to PDE
MATH 22C
1
Introduction To Partial Dierential Equations
Recall: a function:
y = f (t)
t=input=independent variable R (known)
y=output=dependent variable R (unknown)
f=name of function.
Examples: y = t2 , y = et , y = sin t, y = ln t,
2-Motivations From the Theory of ODEs
MATH 22C
1. Ordinary Dierential Equations (ODE) and the
Fundamental Role of the Derivative in the
Sciences
Recall that a real valued function of a real variable y = f (t)
gives outputs y in terms of inputs t.
t=input=
3-EXISTENCE THEOREMS for ODEs
MATH 22C
In this section we consider the initial value problem (ivp)
for rst order autonomous systems of ODEs of the form
y = f (y ),
y(t0) = y0 .
(1)
(2)
Here y(t) = (x(t), y (t) denotes an unknown curve in the
plane paramet
4-THE LINEAR WAVE EQUATION
MATH 22C
In previous sections we considered the linear advection equation
ut + cux = 0,
the equation that describes wave propagation to the right
at speed c. The point is that
u(x, t) = f (x ct)
solves the equation because
ut =