文档介绍：Introduction to Partial Differential Equations and Fourier
Theory
Marcus Pivato
Department of Mathematics
Trent University
Peterborough, Ontario, Canada
December 27, 2003
ii Colour Plates
Colour Plates
(A) (B)
Nairobi
Cairo
Bombay R
Marakesh Sidney Paris
Pisa
Buenos Aires Beijing Kuala Lampur
3
Barcelona Berlin
Copenhagen R
Toronto
Vancouver
Montreal St. Petersburg
Edmonton Halifax Vladivostok
Peterborough Kyoto
New York Santiago C
K
T
S A R
M E N P D F B
Z Q
T V H L
Fig. on page 1: (A) f(C) is the first letter of city C. (B) p(t) is the position of the fly at time t.
r
)
θ z = x + y i
= r [cos(θ) + i sin(θ)]
sin(
r = r cis θ
= θ
y
x= r cos(θ)
p
Fig. on page 5: z = x + yi; r = x2 + y2, θ= tan(y/x).
6+5i
z = 2 + 3 i
2
z = 4 + 2
i 2
1
= 5
2
1
2
y = 3
y = 2
y
+
x = 4 1 x = 2
1 2
y
x+x = 6
1 2
Fig. on page 5: The addition plex numbers z1 = x1 + y1i and z2 = x2 + y2i.
z = 3 cis 75
r 1= 2 z = 2 cis 30
1 z =() cis 45
r=3 1
r=
θ= 30 2 θ= 45
1 2
θ= 75
Colour Plates iii
Fig. on page 5: The multiplication plex numbers z1 = r1 cis θ1 and z2 = r2 cis θ2.
z = 1/2 + π/3 i
e 1/2
exp exp(z) = e 1/2 cis π/3
π= e1/2 [cos( π/3 ) + i sin( π/3 )]
1/2
y= /3 π/3 = e [ 3 /2 + i /2]
x=1/2
Fig. on page 5: The exponential plex number z = x + yi.
Fig. on page 59: If we ‘glue’ the opposite edges of a square together, we get a torus.
f(x)
0
wL(x,t)
-1-t 1-t -1 1
x x+t
t
t
t
Fig. on page 94: The travelling box wave w (x, t) = f (x + t).
L 0
f (w) f (x)
1 1
f (x)
f (w) 2
2
f (w) f (x)
3 3
f (w) f (x)
4 f (z)
4 1 f (z)
2
f (z)
3 f (z)
4
y
w x z
f (x)
f (x) 4
3 f (x)
f (x) 2
1
Fig. on page 125: The sequence {f1, f2, f3, . . .} converges pointwise to the constant 0 function. Thus, if we pick some random
points w, x, y, z ∈ X, then we