Appendix: Orthogonal Curvilinear Coordinates Notes: Most of the material presented in this chapter is taken from Anupam G (Classical Electromagnetism in a Nutshell 2012 (Princeton: New Jersey)) Chap 2 and Weinberg S (Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity 1972 (Wiley: New York) Chap 8 We define the infinitesimal spatial displacement vector dx in a given orthogonal coordinate system with dx = dx i e i (II1) where the Einstein summation convention was used dx i is a contravariant component and e i is a basis vector (i = 123) The length interval ds is thus given by ds 2 = dx dx = ( dx i e i ) dx j e j = ( e i e j )dx i dx j = g ij dx i dx j = dx i dx i (II2) where the orthogonality of the coordinate system is specified by e i e j = g ij with the metric tensor g ij = 0 when i j and dx i is the covariant component Please note that basis vectors are not unit vectors ie e i e i 1 in general Equation (II2) can be used to similarly define the inner product between any two vectors with A B = g ij A i B j = A i B i (II3) Since the covariant and contravariant components are generally different from one another in non-cartesian coordinate systems it is often more desirable to introduce a new set of so-called ordinary or physical components that preserve the inner product without explicitly bringing in both types of components or the metric tensor We start by rewriting equation (II2) as ds 2 = ( du 1 ) 2 + ( du 2 ) 2 + ( du 3 ) 2 (II4) I
for the orthogonal coordinate system ( u 1 u 2 u 3 ) A comparison with equation (II2) reveals that i = g ii For example Cartesian coordinates have = = = 1 cylindrical coordinates ( ρθ z) have = = 1 = ρ and spherical coordinates rθφ have = 1 = r and = rsin( θ ) Going back to equation (II3) for the inner product we now define the physical coordinates A i of a vector A such that A B A i B i (II5) where the use of subscripts has no particular meaning (ie a subscript does not imply a covariant component) A comparison with equation (II3) implies that the physical components are related to the covariant and contravariant components through A i = h i A i = h i 1 A i (II6) The first thing we should notice is that the physical components allow the use of a unit basis ê i since e i e j = g ij = h i h j δ ij = h i h j ê i ê j = ( h i ê i ) ( h j ê j ) (II7) In fact we could have alternatively justified the introduction of the physical components by the desire to use a unit basis with or in general dx = du 1 ê 1 + du 2 ê 2 + du 3 ê 3 = dx 1 ê 1 + dx 2 ê 2 + dx 3 ê 3 (II8) A = A 1 ê 1 + A 2 ê 2 + A 3 ê 3 (II9) It should now be clear that what we usually specify as coordinates (eg ( ρθ z) and ( rθφ) ) correspond to the contravariant components of dx while the physical coordinates are those for which the components of dx have units of length (eg dρ ρdθdz dφ and ( drrdθr sin θ ) ) We now define the different differential operators using the physical coordinates starting with the gradient To do so we first consider the differential of a scalar function f II
df = f u i dui f dx = f h i du i ê i i = h i du i f ê i i (II10) which from the first and last equations implies that f = 1 f u 1 ê1 + 1 f u 2 ê2 + 1 f u 3 ê3 (II11) This leads to the following relations for the cylindrical and spherical coordinate systems f = f ρ êρ + 1 f ρ θ êθ + f z êz f = f r êr + 1 f r θ êθ + 1 f rsin( θ ) φ êφ (II12) respectively For the divergence of a vector we consider an infinitesimal cube as shown in Figure 1 and use the divergence theorem Top e du 3 du 2 Bottom du 1 Figure 1 - Infinitesimal volume of integration where we do not differentiate between and III
Ad 3 x = A du 1 du 2 du 3 V = A nda S (II13) which for a small enough cube we can write as A nda = A 1 S left right top + A 3 bottom du 1 du 2 = A u 1 1 + du 2 du 3 front + A 2 back du 1 du 3 + u 2 A 2 u 3 A 3 du1 du 2 du 3 (II14) since in general h i can vary across the dimensions of the cube A comparison with equation (II13) reveals that 1 A = A u 1 1 + + u 2 A 2 u 3 A 3 (II15) We then respectively have for cylindrical and spherical coordinates A = 1 ρ ρ ρa r A = 1 r 2 + 1 ρ ( r r 2 A r ) + A θ θ + A z z 1 rsin θ θ sin( θ )A θ + 1 A φ rsin( θ ) φ (II16) The Laplacian is readily evaluated by setting A = f and inserting equations (II12) in equations (II16) We then have the corresponding relations 2 f = 1 ρ ρ ρ f ρ + 1 2 f ρ 2 θ + 2 f 2 z 2 2 f = 1 r 2 r r f 2 r + 1 r 2 sin θ θ sin( θ ) f 1 θ + r 2 sin 2 θ 2 f φ 2 (II17) for cylindrical and spherical coordinates respectively Finally for the curl we use Stokes Theorem using an infinitesimal surface as shown in Figure 2 ( A) nda = ( A) ( α du 2 du 3 ê 1 + β du 1 du 3 ê 2 + γ du 1 du 2 ê 3 ) S =! A dl C (II18) IV
e 2 du 2 u 2 = constant e 1 A du 1 B u 1 = constant Figure 2 Infinitesimal loop of integration for the derivation of the curl projection in the where n = αê 1 + βê 2 + γ ê 3 For this infinitesimal loop we can consider the different projections on the three ( u i u j )-planes and write (using the first two terms of the corresponding Taylor expansions)! = α A 2 A dl C bottom top du 2 + front { A 3 back du } 3 top bottom du 1 + right { A 3 left du } 3 { back front du 1 + A 2 left right du } 2 +β A 1 +γ A 1 = α u 3 A 2 +β u 3 A 1 +γ u 2 A 1 + + Equating equations (II18) and (II19) we must have u 2 A 3 u 2 A 3 u 1 A 2 (II19) A = 1 u 2 A 3 u 1 A 2 u 3 A 2 u 2 A 1 A ê1 u 3 1 ê3 u 1 A 3 ê2 (II20) We then respectively write for the cylindrical and spherical coordinate systems V
A = A = 1 A z ρ 1 rsin θ θ A θ z r r ra θ + A ρ êρ z A z êθ ρ ρ ρ ρa θ A θ θ sin( θ )A φ A r êφ θ 1 + φ êr rsin θ A ρ êz θ A r φ 1 r r ra φ êθ (II21) VI