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

304 Chap. 9 • EUCLIDEAN MANIFOLDS
for all
v ∈V . To obtain (43.15) replace v by τ v , τ > 0 in (43.12) and write the result as
f( x+ τ v) − f( x) o( x, τ v )
Av
x
= −
(43.16)
τ
τ
By (43.13) the limit of the last term is zero as τ → 0 , and (43.15) is obtained. Equation (43.15)
holds for all v ∈V because we can always choose τ in (43.16) small enough to ensure that x+
τ v
is in U , the domain of f . If f is differentiable at every x ∈U , then (43.15) can be written
d
grad ( x) v = f( x+
τ v )
(43.17)
d
( f )
τ τ = 0
A function f : U →R, where U is an open subset of E , is called a scalar field. Similarly,
f : U →V is a vector field, and f : U → T ( V ) is a tensor field of order q . It should be noted
that the term field is defined here is not the same as that in Section 7.
q
Before closing this section there is an important theorem which needs to be recorded for
later use. We shall not prove this theorem here, but we assume that the reader is familiar with the
result known as the inverse mapping theorem in multivariable calculus.
r
Theorem 43.7. Let f : U →E ' be a C mapping and assume that grad f ( x
0)
is a linear
isomorphism. Then there exists a neighborhood U
1
of x
0
such that the restriction of f to U
1
is a
r
C diffeomorphism. In addition
−
( ) 1
x x (43.18)
−1
grad f ( f( 0)) = grad f( 0)
This theorem provides a condition under which one can asert the existence of a local inverse of a
smooth mapping.
Exercises
43.1 Let a sequence { x n } converge to x . Show that every subsequence of { n }
converges to x .
x also