ABSTRACT
Vectors in R2 or R3 are physical quantities that have norm (magnitude) and direction. Examples include force, velocity and acceleration. The vectors will be denoted by boldface letters, x, y.
There is a convenient way to express a three dimensional vector x in terms of its components. If
i =
0@ 10 0
1A ; j = 0@ 01 0
1A ; k = 0@ 00 1
1A are the three unit vectors in the Euclidean space R3, then any vector
x =
0@x1x2 x3
1A in R3 can be expressed in the form
x = x1i+ x2j+ x3j:
The norm, magnitude of a vector x = x1i+ x2j+ x3j is dened to be the nonnegative number
kxk = q x21 + x
2 2 + x
Addition of two vectors x = x1i + x2j + x3j and y = y1i + y2j + y3j is dened by
x+ y = x1 + y1
i+
x2 + y2
j+
x3 + y3
j:
Multiplication of a vector x = x1i + x2j + x3j with a scalar c is dened by
c x = cx1i+ cx2j+ cx3j:
The dot product of two vectors x and y in R2 or R3 is dened by
(E.1) x y = kxk kyk cos
where is the angle between the vectors x and y. If x = x1i + x2j + x3j and y = y1i+ y2j+ y3j, then
x y = x1y1 + x2y2 + x3y3:
The cross product of two vectors x and y R3 is dened to be the vector
the unit perpendicular to both vectors x and y and whose direction is given by the right hand side rule. If x = x1i+ x2j+ x3j and y = y1i+ y2j+ y3j, then
(E.3) x y = i j k x1 x2 x3 y1 y2 y3
= x2 x3y2 y3
i x1 x3y1 y3 j+ x1 x2y1 y2
k: A vector-valued function or a vector function is a function of one or more
variables whose range is a set of two or three dimensional vectors. We can write a vector-valued function r = r(t) of a variable t as
r(t) = f(t)i+ g(t)j+ h(t)k:
The derivative r0(t) is given by
r0(t) = f 0(t)i+ g0(t)j+ h0(t)k;
and it gives the tangent vector to the curve r(t). We can write a vector function F(x; y; z) of several variables x, y and z
as F(x; y; z) = f(x; y; z) i+ g(x; y; z) j+ h(x; y; z)k:
A vector function F(x; y; z) usually is called a vector eld. The Laplace operator is one of the most important operators. For a given
function u(x; y), the function
(E.4) r2 u u = uxx + uyy is called the two dimensional Laplace operator or simply two dimensional Laplacian.
