Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
A(u) is said to be continuous at u ˆ u 0 if it is de®ned in some neighborhood of
u 0 and
lim A…u† ˆA…u 0 †:
u!u 0
…1:32†
Note that A(u) is continuous at u 0 if and only if its three components are continuous
at u 0 .
A(u) is said to be di€erentiable at a point u if the limit
dA…u†
du
A…u ‡ u†A…u†
ˆ lim
u!0 u
…1:33†
exists. The vector A 0 …u† ˆdA…u†=du is called the derivative of A(u); and to di€erentiate
a vector function we di€erentiate each component separately:
A 0 …u† ˆA 0 1…u†^e 1 ‡ A 0 2…u†^e 2 ‡ A 0 3…u†^e 3 :
…1:33a†
Note that the unit coordinate vectors are ®xed in space. Higher derivatives of A(u)
can be similarly de®ned.
If A is a vector depending on more than one scalar variable, say u, v for
example, we write A ˆ A…u; v†. Then
is the di€erential of A, and
dA ˆ…@A=@u†du ‡…@A=@v†dv
@A
@u ˆ lim A…u ‡ u; v†A…u; v†
u!0 @u
…1:34†
…1:34a†
and similarly for @A=@v.
Derivatives of products obey rules similar to those for scalar functions.
However, when cross products are involved the order may be important.
Space curves
As an application of vector di€erentiation, let us consider some basic facts about
curves in space. If A(u) is the position vector r(u) joining the origin of a coordinate
system and any point P…x 1 ; x 2 ; x 3 † in space as shown in Fig. 1.11, then Eq. (1.31)
becomes
r…u† ˆx 1 …u†^e 1 ‡ x 2 …u†^e 2 ‡ x 3 …u†^e 3 :
…1:35†
As u changes, the terminal point P of r describes a curve C in space. Eq. (1.35) is
called a parametric representation of the curve C, and u is the parameter of this
representation. Then
r
u
r…u ‡ u†r…u†
ˆ
u
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