Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSand sov i ; i = ∂vρ∂ρ + ∂vφ∂φ + ∂vz∂z + 1 ρ vρ= 1 ∂ρ ∂ρ (ρvρ )+ ∂vφ∂φ + ∂vz∂z .This result is identical to the expression for the divergence of a vector field in cylindricalpolar coordinates given in section 10.9. This is discussed further in section 26.20. ◭So far we have considered only the covariant derivative of the contravariantcomponents v i of a vector. The corresponding result for the covariant componentsv i may be found in a similar way, by considering the derivative of v = v i e i andusing (26.77) to obtainv i; j = ∂v i∂u j − Γk ijv k . (26.88)Comparing the expressions (26.87) and (26.88) for the covariant derivative ofthe contravariant and covariant components of a vector respectively, we see thatthere are some similarities and some differences. It may help to remember thatthe index with respect to which the covariant derivative is taken (j in this case), isalso the last subscript on the Christoffel symbol; the remaining indices can thenbe arranged in only one way without raising or lowering them. It only remains tonote that for a covariant index (subscript) the Christoffel symbol carries a minussign, whereas for a contravariant index (superscript) the sign is positive.Following a similar procedure to that which led to equation (26.87), we mayobtain expressions for the covariant derivatives of higher-order tensors.◮By considering the derivative of the second-order tensor T with respect to the coordinateu k , find an expression for the covariant derivative T ij ; kof its contravariant components.Expressing T in terms of its contravariant components, we have∂T∂u = ∂k ∂u (T ij e k i ⊗ e j )ij∂T=∂u e k i ⊗ e j + T ij ∂e i∂u ⊗ e k j + T ij e i ⊗ ∂e j∂u . kUsing (26.75), we can rewrite the derivatives of the basis vectors in terms of Christoffelsymbols to obtainij∂T ∂T=∂uk ∂u e k i ⊗ e j + T ij Γ l ike l ⊗ e j + T ij e i ⊗ Γ l jke l .Interchanging the dummy indices i and l in the second term and j and l in the third termon the right-hand side, this becomes( )∂T ∂Tij∂u = +Γ i k ∂ulkT lj +Γ j k lk T il e i ⊗ e j ,970