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

314 Chap. 9 • EUCLIDEAN MANIFOLDS
z
= x
3 3
2
z
1
z
2
x
Figure 9
1
x
Returning to the general transformations (44.5) and (44.6), we can substitute the second into
the first and differentiate the result to find
∂y
∂x
∂x
∂y
i
k
1 N 1 N i
( x ,..., x ) ( y ,..., y ) = δ
k
j
j
(44.28)
By a similar argument with x and y interchanged,
∂y
∂x
∂x
∂y
k
i
1 N 1 N i
( x ,..., x ) ( y ,..., y ) = δ
j
k
j
(44.29)
Each of these equations ensures that
⎡∂y
⎤
i
1 N
det ⎢ ( x ,..., x ) 0
k
j
x
⎥ = ≠
⎣∂ ⎦ ⎡∂x
1 N ⎤
det ( y ,..., y )
l
⎢
⎣∂y
1
⎥
⎦
(44.30)
For example, ( j k
⎡
⎣ )( )
j k 1 N
determinant det ⎡( ∂x ∂y )( y ,..., y ) ⎤
⎤
⎦
1 2 3 1
det ∂z ∂ x x , x , x = x ≠0
⎣ ⎦
for the cylindrical coordinate system. The
is the Jacobian of the coordinate transformation (44.6).