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A.1 General Case (u1,u2,u3)

We consider the curvilinear coordinate system (u1,u2,u3).

A.1.1 Vector Algebra

Covariant (Tangent) Basis

The covariant, or tangent vector basis (e1,e2,e3)is defined as

∂X ei = ∂ui-
(A.1)

where the ei are tangent to the curvilinear lines. They can be normalized by

ei ^ei = h- i
(A.2)

where we introduce the scale factors

||∂X || hi = ||--i|| ∂u
(A.3)

Contravariant (reciprocal) Basis

The gradient ∇f of a function f being defined by the differential

df = ∇f ⋅dX
(A.4)

we apply to ui which gives

dui = ∇ui ⋅dX
(A.5)

By chain rule, we have

∂X j j dX = ∂ujdu = ejdu
(A.6)

so that

i i j du = ∇u ⋅ejdu
(A.7)

which implies

∇ui ⋅e = δi j j
(A.8)

thus defining two reciprocal basis ( i ) ∇u ,ej of vectors. The reciprocal basis vectors are also called contravariant, and noted

i i e ≡ ∇u
(A.9)

These vectors are perpendicular to the surfaces of constant ui.

From the properties of reciprocal basis, we can calculate a vector from the three vectors of the reciprocal basis, such that

ei = ∇ui = --ej-×-ek- ei ⋅ej × ek (A.10)
ei = ∂X-- ∂ui = j k --e-×--e-- ei ⋅ej × ek (A.11)

Metric Coefficients

They are defined as

gij = ei ⋅ ej (A.12)
gij = ei ⋅ ej

With the differential vector given in (A.6), we show that the differential arc length along a curve is

√ ------- ∘ --------- dl = |dX |= dX ⋅dX = gijduiduj
(A.13)

In addition, we have the relations

ei = gijej (A.14)
ei = gije j

We also show that

[gij] = [ ] gij-1 (A.15)
[ ij] g = [gij]-1 (A.16)

so that, defining

g = det[gij]
(A.17)

we find

[ ] g-1 = det gij
(A.18)

Jacobian

We define the Jacobian

( ∂x ∕∂u1 ∂x∕∂u2 ∂x∕∂u3 ) J ≡ --∂(x,y,z)- = det( ∂y ∕∂u1 ∂y∕∂u2 ∂y∕∂u3 ) ∂(u1,u2,u3) 1 2 3 ∂z∕ ∂u ∂z∕∂u ∂z∕∂u
(A.19)

which gives

∂X ∂X ∂X J = --1-⋅--2-× ---3 = e1 ⋅e2 × e3 ∂u ∂u ∂u
(A.20)

and the reciprocal Jacobian

( 1 1 1 ) ∂(u1,u2,u3) ∂u ∕∂x ∂u ∕∂y ∂u ∕∂z J ≡ -∂(x,y,z)--= det( ∂u2∕∂x ∂u2 ∕∂y ∂u2∕∂z ) ∂u3∕∂x ∂u3 ∕∂y ∂u3∕∂z
(A.21)

which gives

1 2 3 1 2 3 J = ∇u ⋅∇u × ∇u = e ⋅e × e
(A.22)

We can show that

- 1 J = J
(A.23)

and the relations (A.10-A.11) become

ei = 1- J(ej × ek) (A.24)
ei = J( ) ej × ek (A.25)

Also,

g = J 2
(A.26)

Vector Identities

With

A = (A ⋅ei)ei = A iei (A.27)
A = ( i) A ⋅ee i = Aie i (A.28)

we find

A ⋅B = gijAiBj = gijAiBj
(A.29)

so that

∘ ------- ∘ -------- A = |A | = gijAiAj = gijAiAj
(A.30)

We also find

i j i j A × B = A B ei × ej = AiBje × e
(A.31)

which gives

i j εijk (A × B)k = εijkJ A B = -J--AiBj
(A.32)

Note that from (A.14),

Ai = gijAj (A.33)
Ai = gijA j (A.34)

Differential elements

differential length along ui

dl(i) = |dX (i)| = hdui = √g-dui i ii
(A.35)

Equivalently,

|| j k|| i dl(i) = J|∇u × ∇u |du
(A.36)

differential area in surface of constant ui Using

j k dS (i) = |dX (j)× dX (k)| = |ej × ek |du du
(A.37)

which becomes

∘ ----------- dS (i) = g g - g2 dujduk jj kk jk
(A.38)

Equivalently

|| i|| j k dS (i) = J ∇u du du
(A.39)

so that

j k i dS (i) = ąJ du du ∇u
(A.40)

differential volume element

d3X =dX (1)⋅dX (2) × dX (3) = J du1du2du3
(A.41)

Vector Differentiation
∂A k dA = ∂uk-du
(A.42)

-∂A- ∂uk = ( ) ∂A-- ∂ukje j ≡ Akje j (A.43)
= ( ) ∂A-- ∂ukjej ≡ A j,kej (A.44)

with

Akj = j ∂A-- ∂uk + Ai{ } ∂ei-⋅ej ∂uk (A.45)
Aj,k = ∂Aj ---k ∂u + Ai{ ∂ei } --k-⋅ej ∂u (A.46)

Then,

δAj = (dA )j = ∂Aj- ∂ukduk + Ai{ } ∂ei- j ∂uk ⋅eduk (A.47)
= dAj + { } ∂eik ⋅ej ∂uAiduk (A.48)

It can be shown that the Christoffel Symbol of the second kind is

{ } { } [ ] -∂ei⋅ej ≡ j = 1-gjn ∂gni + ∂gnk-- ∂gik ∂uk i k 2 ∂uk ∂ui ∂un
(A.49)

so that

Akj = j ∂A-- ∂uk + { } j i kAi (A.50)
Aj,k = ∂A --jk- ∂u -{ i } j kAi (A.51)

Note that since

∂ei-= ∂ek- ∂uk ∂ui
(A.52)

we have

{ } { } j j i k = k i
(A.53)

Operator ∇

The operator ∇ can be decomposed in the curvilinear coordinates as

i-∂-- i-∂-- ∇ = ∇u ∂ui = e ∂ui
(A.54)

We then find the following differential operations:

Gradient It follows simply that

i ∂f- ∂f--i ∇f = ∇u ∂ui = ∂uie
(A.55)

so that

∂f (∇f )i = (∇f ⋅ei) = ∂ui-
(A.56)

Divergence It can be shown that the divergence is expressed as

1--∂-( i) ∇ ⋅A = J ∂ui J A
(A.57)

Curl It becomes, a compact notations,

εijk-∂Aj- ∇ × A = J ∂ui ek
(A.58)

or is extended as

( ) k 1 ∂Aj ∂Ai (∇ × A ) = J- -∂ui - ∂uj-
(A.59)

A.1.2 Tensor Algebra

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