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

330 Chap. 9 • EUCLIDEAN MANIFOLDS
If we select
N
dr = g ( x) dx , dr = g ( x) dx , ⋅⋅⋅ , dr = g ( x ) dx , we can write (45.31)) as
1 2
1 1 2 2
N
N
1 2
( g x g x )
( ), , ( ) N
dυ = E
1
⋅⋅⋅
N
dx dx ⋅⋅⋅dx
(45.33)
By use of (45.26) and (45.27), we then get
( g ( x), , g ( x)
)
E ⋅⋅⋅ = E = e g
(45.34)
1 N
12⋅⋅⋅N
Therefore,
1 2 N
dυ = gdx dx ⋅⋅⋅ dx
(45.35)
For example, in the parabolic coordinates mentioned in Exercise 44.4,
(( ) ( ) )
2 2
d x x x x dx dx dx
1 2 1 2 1 2 3
υ = + (45.36)
Exercises
45.1 Let v be a C ∞ vector field and f be a C ∞ function both defined on U an open set in E .
We define v f : U →R by
v f( x) = v( x) ⋅grad f( x),
x∈U (45.37)
Show that
( λf + μg) = λ( f ) + μ( g)
v v v
and