Page 165 - Schaum's Outline of Theory and Problems of Advanced Calculus

156 VECTORS [CHAP. 7

primary reason for introducing this approach was to formalize a component representation of the
vectors. It is that concept that will be used in the remainder of this chapter.
Note 1: One of the advantages of component representation of vectors is the easy extension of the
ideas to all dimensions. In an n-dimensional space, the component representation is
AðA 1 ; A 2 ; ... ; A n Þ
An exception is the cross-product which is specifi-
cally restricted to three-dimensional space. There are
generalizations of the cross-product to higher dimen-
sional spaces, but there is no direct extension.)
Note 2: The geometric interpretation of a vector
endows it with an absolute meaning at any point of
space. The component representation (as an ordered
triple of numbers) in Euclidean three space is not unique,
rather, it is attached to the coordinate system employed.
This follows because the components are geometrically
interpreted as the projections of the arrow representation
on the coordinate directions. Therefore, the projections
on the axes of a second coordinate system rotated (for
example) from the first one will be different. (See Fig.
7-10.) Therefore, for theories where groups of coordinate
systems play a role, a more adequate component defini-
tion of a vector is as a collection of ordered triples of
numbers, each one identified with a coordinate system
of the group, and any two related by a coordinate
transformation. This viewpoint is indispensable in New-
tonian mechanics, electromagnetic theory, special relativ-
ity, and so on.
Fig. 7-10

VECTOR FUNCTIONS
If corresponding to each value of a scalar u we associate a vector A, then A is called a function of u
denoted by AðuÞ.In three dimensions we can write AðuÞ¼ A 1 ðuÞi þ A 2 ðuÞj þ A 3 ðuÞk.
The function concept is easily extended. Thus, if to each point ðx; y; zÞ there corresponds a vector
A, then A is a function of ðx; y; zÞ, indicated by Aðx; y; zÞ¼ A 1 ðx; y; zÞi þ A 2 ðx; y; zÞj þ A 3 ðx; y; zÞk.
We sometimes say that a vector function A defines a vector field since it associates a vector with each
point of a region. Similarly, ðx; y; zÞ defines a scalar field since it associates a scalar with each point of a
region.

LIMITS, CONTINUITY, AND DERIVATIVES OF VECTOR FUNCTIONS
Limits, continuity, and derivatives of vector functions follow rules similar to those for scalar func-
tions already considered. The following statements show the analogy which exists.
1. The vector function represented by AðuÞ is said to be continuous at u 0 if given any positive
number ,we can find some positive number such that jAðuÞ Aðu 0 Þj < whenever
ju u 0 j < . This is equivalent to the statement lim AðuÞ¼ Aðu 0 Þ.
u!u 0
2. The derivative of AðuÞ is defined as
dA Aðu þ uÞ AðuÞ
¼ lim
du u!0 u
provided this limit exists. In case AðuÞ¼ A 1 ðuÞi þ A 2 ðuÞj þ A 3 ðuÞk; then