Chapter 11. Fourier Analysis
1
11.1 Fourier Series
Denition (Periodic Functions)
A function f (x) is called a periodic function if p > 0
(called a period of f ) such that f (x + p) = f (x) for all x.
Ex. periodic : sin 2x, cos 3x, sin 2x + 3 cos 3x,
not

Chapter 20. Numerical Linear Algebra
1
20.1 Linear Systems: Gauss Elimination
A linear system of n equations in n unknowns
E1 : a11x1 + a12x2 + + a1nxn = b1
E2 : a21x1 + a22x2 + + a2nxn = b2
En : an1x1 + an2x2 + + annxn = bn
A single vector equation
Ax

Chapter 12. Partial Dierential Equations(PDEs)
1
12.1 Basic Concept of PDEs
Partial dierential equations(PDEs)
- an equation involving one or more partial derivatives of an unknown function that depends on two or more variables.
Ex. When u = u(x, y), ux

Chapter 10. Vector Integral Calculus. Integral Theorems
1
10.1 Line integrals
Denition (Line integral)
Let C be a (piecewise smooth) curve
r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k (a t b),
and let F be a vector function over C defined by
F(r) = F

Chapter 9. Vector Dierential Calculus, Grad, Div, Curl
1
9.7 Gradient of a Scalar Field
Definition (Gradient)
Let f (x, y, z) be dierentiable. Then the gradient of
f (x, y, z) is dened
[ by the vector
] function
f
f
f
f f f
,
,
=
i+
j+
k
gradf = f =
x y z

#1.
,
#2. (a)
cos , where
sin
(b) cos sin
(c)
#3.
#4.
#5. Hint :By Stokes theorem,
.
So we only need to show that .
#6. (a) 11.9 #3(a)
(b)
#7. (a) cos cos cos cos
(b)
#8. (a)
sin cos sin

1
:
(4)
:
:
1. (5pts) Find a unit vector in the direction in
which decreases most rapidly
(b) (2pts) For arbitrary point in , find the
normal vector at .
at , and find the rate of change of
at in that direction.
(c) (4pts) Find the area of the surfac

Homework
23.1
1. Find the adjacency matrix of the given graph or digraph.
(a)
(b)
(c)
1
(d)
2. Incidence matrix
cfw_ B of a graph is B = [bjk ], where
1 if vertex j is an endpoint of edge ek
bjk =
0 otherwise
of a digraph is B
= [bjk ], where
Incidence

Homework
1
12.1
1. Verify that u(x, t) = v(x + ct) + w(x ct) with any twice dierentiable function
2
2u
2 u
.
v and w is a solution of 1-dim wave equation 2 = c
t
x2
2. Verify that the function u(x, y) = a ln(x2 +y 2 )+b satises 2-dim Laplace equation
and

Homework
1
11.1
1. (a) If f (x) and g(x) have period p, show that h(x) = af (x) + bg(x)(a, b, constant)
has the period p.
(b) If f (x) has period p, show that f (bx), b = 0, is a periodic function of x of period
p/b.
2. Find the Fourier series of the give

Homework
1
9.8 9.9
1. Find the divergence and the curl of the following vector functions.
(a) F(x, y, z) = ex i+yex j+2z sinh xk (b) F(x, y, z) = (x2 +y 2 +z 2 )3/2 (xi+yj+zk)
2. Show that the flow with velocity vector v = yi is incompressible. Show that

Chapter 23. Graphs. Combinatorial Optimization
1
23.1 Graphs and Digraphs
Denition (Graph)
A graph G consists of two nite sets (sets having nitely many elements),
a set V of points, called vertices, and a set E of connecting lines, called
edges, such that