Introduction to Vectors and Tensors Vol 2 (Bowen 246). - Index of

324 Chap. 9 • EUCLIDEAN MANIFOLDS
Section 45. Transformation Rules for Vectors and Tensor Fields
In this section, we shall formalize certain ideas regarding fields on E and then investigate
the transformation rules for vectors and tensor fields. Let U be an open subset of E ; we shall
denote by F ∞ ( U ) the set of C ∞ functions f : U →R. First we shall study the algebraic structure
of F ∞ ( U ) . If f and 1
f
2
are in F ∞ ( U ) , then their sum f1+ f2
is an element of F ∞ ( U ) defined by
( f + f )( x) = f ( x) + f ( x )
(45.1)
1 2 1 2
and their produce f1f 2
is also an element of F ∞ ( U ) defined by
( ff)( x) = f( x) f( x )
(45.2)
1 2 1 2
for all
x ∈U . For any real number λ ∈R the constant function is defined by
λ( x ) = λ
(45.3)
for all x ∈U . For simplicity, the function and the value in (45.3) are indicated by the same
symbol. Thus, the zero function in F ∞ ( U ) is denoted simply by 0 and for every f ∈ F ∞ ( U )
f
+ 0= f
(45.4)
It is also apparent that
1f
= f
(45.5)
In addition, we define
− f = ( − 1) f
(45.6)