Appendix B Introduction to Curvilinear Coordinates B.1 Definition of a Vector A vector, v, in three-dimensional space is represented in the most general form as the summation of three components, v1, v2 and v3, aligned with three “base” vectors, as follows: 3 1 2 3 i v = v g1 + v g2 + v g3 = ∑ v gi (B.1) i=1 where bold typeface denotes vector quantities and the base vectors, gi, can be non-orthogonal and do not have to be unit vectors as long as they are non-coplanar. The subscript i indicates a covariant quantity and the superscript i indicates a contravariant quantity, hence the above formula describes vector v as three contravariant components of the covariant base vectors. The Einstein summation convention only applies where one dummy index i is subscript and the other is superscript (summation does not apply over a repeated subscript i, so that for instance the metric tensor gii, discussed later, has 3 separate components). B.2 Transformation Properties of Covariant and Contravariant Ten- sors The subject of covariant and contravariant tensors is often introduced in tensor analysis text books by defining the behaviour of the two under transformation. The gradient of a scalar, φ, is given by the following expression in general non-orthogonal coordinates (ξ;η;ζ): ∂φ ∂φ ∂φ ∂φ ∇φ = g1 + g2 + g2 = gi (B.2) ∂ξ ∂η ∂ζ ∂ξi 155 156 APPENDIX B. Introduction to Curvilinear Coordinates If one defines another coordinate system ξ;η;ζ then components of the gradient can be expressed using the chain-rule: ∂φ ∂ξ j ∂φ = (B.3) ∂ξi ∂ξi ∂ξ j which can be written: j Ai = ai A j (B.4) where: j ∂φ j ∂ξ ∂φ Ai = a = A j = (B.5) ∂ξ j i ∂ξi ∂ξ j Tensors that satisfy this transformation are called covariant tensors and have lowered subscripts, as in Ai. To examine the transformation properties of a contravariant tensor, the vector dr is considered, as follows: dr = dξg1 + dηg2 + dζg3 (B.6) As before, if one defines another coordinate system ξ;η;ζ then components of the vector can be expressed using the chain-rule: ∂ξi dξi = dξ j (B.7) ∂ξ j This can be written: i i j A = b jA (B.8) where: ∂ξi Ai dξi bi = A j = dξ j (B.9) ≡ j ∂ξ j Tensors that transform according to Equation (B.8) are termed contravariant, and have raised indices. i B.3 Covariant and Contravariant Base Vectors, gi and g One can define a point in space by the position vector, r, using the familiar Cartesian coordinates, as follows: r = xˆi + yjˆ + zkˆ 1 2 3 = x e1 + x e2 + x e3 i = x ei (B.10) and, equally, one can define the unit vector in the x-direction, ˆi, as follows: ∂r ˆi = (B.11) ∂x i B.3. Covariant and Contravariant Base Vectors, gi and g 157 or, more generally: ∂r e = (B.12) i ∂xi The same point in space can be defined using a more general coordinate system: r = ξg1 + ηg2 + ζg3 1 2 3 = ξ g1 + ξ g2 + ξ g3 i = ξ gi (B.13) where: ∂r g = (B.14) i ∂ξi Equations (B.10) and (B.13) are equivalent. Using the chain rule, one can therefore express the covariant general base vectors gi in terms of the covariant Cartesian base vectors, ei, as follows: ∂r ∂r ∂ξ j = ∂xi ∂ξ j ∂xi ∂ξ j e = g (B.15) i ∂xi j and likewise: ∂r ∂r ∂x j = ∂ξi ∂x j ∂ξi ∂x j g = e (B.16) i ∂ξi j The covariant and contravariant base vectors are defined such that the scalar product of the covari- ant and contravariant base vectors is unity, i.e.: j gi g = 1 if i = j · = 0 if i = j 6 or: j j gi g = δ (B.17) · i j i j where δ δ δi j is the Kronecker delta. i ≡ ≡ In Equation (B.13), the vector r was expressed in terms of the covariant base vector gi. In a similar way, vector r can be written in terms of the contravariant base vector gi: i r = ξig (B.18) 158 APPENDIX B. Introduction to Curvilinear Coordinates where, following a similar analysis to that given for Equation (B.16): ∂r ∂r ∂x = j (B.19) ∂ξi ∂x j ∂ξi i j and since g = ∂r=∂ξi and e = ∂r=∂x j, the contravariant base vector is given by: ∂x gi = j e j (B.20) ∂ξi One can obtain the covariant and contravariant components from the scalar product of the vector, i r, and the corresponding base vectors (gi or g ), as follows: j j r gi = ξ jg gi = ξ jδ = ξi (B.21) · · i i j i j i i r g = ξ g j g = ξ δ = ξ (B.22) · · j δ j ξ ξ ξ j where i has substitution operator properties (i.e. it changes the component j to i, or from to ξi). Comparing Equations (B.18) and (B.21) one can also see that if the base vector is taken from the i right-hand-side to the left-hand-side of Equation (B.18), the superscript g becomes subscript gi. There is an alternative method to obtaining the contravariant base vector gi as a function of e j to that shown above. Returning to Equation (B.15), it was shown that: ∂ξ j e = g (B.23) k ∂xk j Taking the scalar product of both sides of this equation with gi: ∂ξ j ∂ξ j ∂ξi i i i ek g = g j g = δ = (B.24) · ∂xk · ∂xk j ∂xk Now, assuming that the contravariant base vector gi can be obtained from e j using a linear combination αi of factors j: i αi 1 αi 2 αi 3 αi j g = 1e + 2e + 3e = je (B.25) and taking the scalar product of both sides of this with ek: i i j i j i g ek = α e ek = α δ = α (B.26) · j · j k k i i k where g ek = ∂ξ =∂x from Equation (B.24) and: · ∂ξi αi = (B.27) j ∂x j i B.3. Covariant and Contravariant Base Vectors, gi and g 159 Finally, from Equation (B.25), one obtains: ∂ξi gi = e j (B.28) ∂x j which is the same result as Equation (B.20). 1 The vector product (g2 g3) has magnitude equal to the area of the rectangle with sides g2 and × g3, with direction nˆ normal to both g2 and g3. The scalar product (g1 nˆ) is equivalent to a distance in · the normal direction, thus the volume of the parallelepiped spanned by vectors g1, g2 and g3 is given by: ∆Vol = g1 (g2 g3) (B.30) · × The contravariant base vectors also satisfy: 1 1 2 1 3 1 g = (g2 g3) g = (g3 g1) g = (g1 g2) (B.31) ∆Vol × ∆Vol × ∆Vol × and similarly the covariant base vectors satisfy: 1 2 3 1 3 1 1 1 2 g1 = ∆ g g g2 = ∆ g g g3 = ∆ g g (B.32) Vol0 × Vol0 × Vol0 × 1 2 3 where ∆Vol0 = g g g represents the volume of the parallelepiped spanned by the contravariant 1 2· ×3 base vectors g , g and g . It is useful to note at this point that the covariant and contravariant rectangular Cartesian base vec- m tors are identical, e em. This is partly why covariant and contravariant tensors are not mentioned in ≡ most fluid mechanics text books which only deal with Cartesian tensors. The equivalence of covariant and contravariant Cartesian tensors is demonstrated by: 1 1 g = (g2 g3) (B.33) ∆Vol × which states that the contravariant g1 vector is perpendicular to the plane defined by the two covariant 1 vectors, g2 and g3. In Cartesian coordinates there is no distinction between g and g1 since the kˆ vector is orthogonal to the plane defined by the ˆi and jˆ vectors (i.e. the g1 vector is perpendicular to the plane defined by g2 and g3). 1The vector product is defined as: (g2 g3) = ( g2 g3 sinθ)nˆ (B.29) × j jj j where nˆ is the unit normal to vectors g2 and g3 and θ is the angle between the two g2 and g3 vectors. Since the area of a triangle with sides g2 and g3 is determined from (1=2 base height) which is equivalent to (1=2 g2 g3 sinθ), the × × × j j × j j magnitude of the cross product must be equal to the area of the rectangle with sides g2 and g3 (i.e. two triangles). 160 APPENDIX B. Introduction to Curvilinear Coordinates B.4 The Jacobian Matrix, [J] It has previously been shown (Equations B.16 and B.28) that the covariant and contravariant base i j vectors, gi and g , can be expressed in terms of the Cartesian base vectors, e j or e , as follows: ∂x j g = e (B.34) i ∂ξi j ∂ξi gi = e j (B.35) ∂x j The Jacobian matrix, [J], is defined as the matrix of coefficients appearing in Equation (B.34): xξ xη xζ ∂x j [J] = = η (B.36) ∂ξi 2 yξ y yζ 3 zξ zη zζ 6 7 4 5 where, for example, xξ ∂x=∂ξ and all components are contravariant, i.e.: ≡ x x1 ≡ y x2 ≡ z x3 ≡ ξ ξ1 ≡ η ξ2 ≡ ζ ξ3 ≡ B.5 Determinant of the Jacobian Matrix, J The Jacobian, J, is defined as the determinant of the Jacobian matrix: J = det[J] = xξ yηzζ yζzη xη yξzζ yζzξ + xζ yξzη yηzξ (B.37) − − − − It was noted earlier that the base vectors used to describe vector r in three-dimensional space should not be coplanar.