Mathematical Methods for Physics and Engineering - Matematica.NET

26.19 COVARIANT DIFFERENTIATIONconstant (this term vanishes in Cartesian coordinates). Using (26.75) we write∂v∂u j = ∂vi∂u j e i + v i Γ k ije k .Since i and k are dummy indices in the last term on the right-hand side, we mayinterchange them to obtain( )∂v∂u j = ∂vi∂v∂u j e i + v k Γ i ikje i =∂u j + vk Γ i kj e i . (26.86)The reason for the interchanging the dummy indices, as shown in (26.86), is thatwe may now factor out e i . The quantity in parentheses is called the covariantderivative, for which the standard notation isv i ; j ≡ ∂vi∂u j +Γi kjv k , (26.87)the semicolon subscript denoting covariant differentiation. A similar short-handnotation also exists for the partial derivatives, a comma being used for theseinstead of a semicolon; for example, ∂v i /∂u j is denoted by v i ,j. In Cartesiancoordinates all the Γ i kjare zero, and so the covariant derivative reduces to thesimple partial derivative ∂v i /∂u j .Using the short-hand semicolon notation, the derivative of a vector may bewritten in the very compact form∂v∂u j = vi ; je iand, by the quotient rule (section 26.7), it is clear that the v i ; j are the (mixed)components of a second-order tensor. This may also be verified directly, usingthe transformation properties of ∂v i /∂u j and Γ i kjgiven in (26.84) and (26.78)respectively.In general, we may regard the v i ; j as the mixed components of a secondordertensor called the covariant derivative of v and denoted by ∇v. In Cartesiancoordinates, the components of this tensor are just ∂v i /∂x j .◮Calculate v i ; i in cylindrical polar coordinates.Contracting (26.87) we obtainNow from (26.83) we havev i ; i = ∂vi∂u i +Γ i kiv k .Γ i 1i =Γ 1 11 +Γ 2 12 +Γ 3 13 =1/ρ,Γ i 2i =Γ 1 21 +Γ 2 22 +Γ 3 23 =0,Γ i 3i =Γ 1 31 +Γ 2 32 +Γ 3 33 =0,969