A short introduction to complex numbers
We dene i to be the square root of 1, that is, i2 = 1, or i =
We can then let
C = cfw_a + bi | a, b real numbers
1.
We also dene the sum and product of z1 = a1 + b1 i and z2 = a2 + b2 i.
addition: z1 + z2 = (a1 + b
The Laplacian in Polar Coordinates
Let u = u (x, y).
The Laplacian is 2 u = u =
2u
x2
+
2u
y2
.
We now find an expression for 2 u in polar coordinates.
Let x = r cos () and y = r sin () ; r > 0.
These define r and implicitly as functions of x and y. In
Partial Differential Equations and Boundary and Initial Conditions
Section A: Standard Partial Differential Equations and
Boundary and Initial Conditions for Problems in Two Variables
I.
The 1-D Wave Equation: c 2 u xx = u tt ; 0 < x < L , t > 0 ; (c and
Partial Differential Equations and Boundary & Initial Conditions
I.
The 1-D Wave Equation: c2 u xx = u tt ; 0 < x < L, t > 0; (c, L > 0).
Figure 1
Consider the transverse vibrations of a string of length L (see Figure 1). (We assume that
there is no dampi