Page 308 - Intro to Tensor Calculus

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The transformations (2.5.62) and (2.5.64) are inverses of each other and so we can write

    −1
ξ x ξ y ξ z x ξ x η x ζ
 η x η y η z  =  y ξ y η y ζ 
ζ x ζ y ζ z z ξ z η z ζ
(2.5.65)
 
y η z ζ − y ζ z η −(x η z ζ − x ζ z η ) x η y ζ − x ζ y η
=J  −(y ξ z ζ − y ζ z ξ ) x ξ z ζ − x ζ z ξ −(x ξ y ζ − x ζ y ξ ) 
−(x ξ z η − x η z ξ )
y ξ z η − y η z ξ x ξ y η − x η y ξ
By comparing like elements in equation (2.5.65) we obtain the relations
ξ x =J(y η z ζ − y ζ z η ) η x = − J(y ξ z ζ − y ζ z ξ ) ζ x =J(y ξ z η − y η z ξ )

ξ y = − J(x η z ζ − x ζ z η ) η y =J(x ξ z ζ − z ζ z ξ ) ζ y = − J(x ξ z η − x η z ξ ) (2.5.66)
ξ z =J(x η y ζ − x ζ y η ) η z = − J(x ξ y ζ − x ζ y ξ ) ζ z =J(x ξ y η − x η y ξ )

The equations (2.5.55) can now be written in terms of the new variables (ξ, η, ζ)as

∂U ∂E ∂E ∂E ∂F ∂F ∂F ∂G ∂G ∂G
+ ξ x + η x + ζ x + ξ y + η y + ζ y + ξ z + η z + ζ z =0 (2.5.67)
∂t ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ
Now divide each term by the Jacobian J and write the equation (2.5.67) in the form

∂ U ∂ Eξ x + Fξ y + Gξ z
+
∂t J ∂ξ J

∂ Eη x + Fη y + Gη z
+
∂η J

∂ Eζ x + Fζ y + Gζ z
+
∂ζ J
(2.5.68)

∂ ξ x ∂ η x ∂ ζ x
− E + +
∂ξ J ∂η J ∂ζ J

∂ ξ y ∂ η y ∂ ζ y
− F + +
∂ξ J ∂η J ∂ζ J

∂ ξ z ∂ η z ∂ ζ z
− G + + =0
∂ξ J ∂η J ∂ζ J
Using the relations given in equation (2.5.66) one can show that the curly bracketed terms above are all zero
and so the transformed equations (2.5.55) can also be written in the conservative form
∂U ∂E ∂F b ∂G
b
b
b
+ + + =0 (2.5.69)
∂t ∂ξ ∂η ∂ζ
where
U
U =
b
J
Eξ x + Fξ y + Gξz
E =
b
J
(2.5.70)
Eη x + Fη y + Gη z
F =
b
J
Eζ x + Fζ y + Gζ z
G =
b
J