348 APPENDIX B. CURVILINEAR COORDINATES

U

D D̂

M N

(x1,x2,x3) (z1,z2,z3)

Thus M (x) = N (z) and so z = N−1 (M (x)) . The point in U will be denoted inrectangular coordinates as y and we have y (x) = y (z) Now by the chain rule,

ei (z) =∂y

∂ zi =∂y

∂x j∂x j

∂ zi=

∂x j

∂ zi e j (x) (2.19)

Define the covariant and contravariant coordinates for the various curvilinear coordinatesin the obvious way. Thus,

v = vi (x)ei (x) = vi (x)ei (x) = v j (z)e

j (z) = v j (z)e j (z) .

Then the following theorem tells how to transform the vectors and coordinates.

Theorem B.5.1 The following transformation rules hold for pairs of curvilinearcoordinates.

vi (z) =∂x j

∂ ziv j (x) , vi (z) =

∂ zi

∂x j v j (x) , (2.20)

ei (z) =∂x j

∂ zie j (x) , e

i (z) =∂ zi

∂x j ej (x) , (2.21)

gi j (z) =∂xr

∂ zi∂xs

∂ z j grs (x) , gi j (z) =∂ zi

∂xr∂ z j

∂xs grs (x) . (2.22)

Proof: We already have shown the first part of 2.21 in 2.19. Then, from 2.19,

ei (z) = ei (z) ·e j (x)ej (x) = ei (z) · ∂ zk

∂x j ek (z)ej (x)

= δik

∂ zk

∂x j ej (x) =

∂ zi

∂x j ej (x)

and this proves the second part of 2.21. Now to show 2.20,

vi (z) = v ·ei (z) = v·∂x j

∂ zie j (x) =

∂x j

∂ ziv ·e j (x) =

∂x j

∂ ziv j (x)

and

vi (z) = v ·ei (z) = v · ∂ zi

∂x j ej (x) =

∂ zi

∂x j v ·ej (x) =

∂ zi

∂x j v j (x) .

To verify 2.22,

gi j (z) = ei (z) ·e j (z) = er (x)∂xr

∂ zi ·es (x)∂xs

∂ z j = grs (x)∂xr

∂ zi∂xs

∂ z j . ■