Mathematical Methods for Physics and Engineering - Matematica.NET

26.16 GENERAL COORDINATE TRANSFORMATIONS AND TENSORSso that the basis vectors in the old and new coordinate systems are related bye j = ∂u′ i∂u j e′ i. (26.67)Now, since we can write any arbitrary vector a in terms of either basis asa = a ′i e ′ i = a j e j = a j ∂u′i∂u j e′ i,it follows that the contravariant components of a vector must transform asa ′i = ∂u′ i∂u j a j . (26.68)In fact, we use this relation as the defining property for a set of quantities a i toform the contravariant components of a vector.◮Find an expression analogous to (26.67) relating the basis vectors e i and e ′i in the twocoordinate systems. Hence deduce the way in which the covariant components of a vectorchange under a coordinate transformation.If we consider the second set of basis vectors in (26.66), e ′i = ∇u ′i , we have from the chainrule that∂u j∂x = ∂u j ∂u ′i∂u ′i ∂xand similarly for ∂u j /∂y and ∂u j /∂z. So the basis vectors in the old and new coordinatesystems are related bye j = ∂u j∂u ′i e′i . (26.69)For any arbitrary vector a,a = a ′ ie ′i = a j e j = a j∂u je′i∂u′iand so the covariant components of a vector must transform asa ′ i = ∂u j∂u a j. (26.70)′iAnalogously to the contravariant case (26.68), we take this result as the defining propertyof the covariant components of a vector. ◭We may compare the transformation laws (26.68) and (26.70) with those fora first-order Cartesian tensor under a rigid rotation of axes. Let us considera rotation of Cartesian axes x i throughanangleθ about the 3-axis to a newset x ′i , i =1, 2, 3, as given by (26.7) and the inverse transformation (26.8). It isstraightforward to show that∂x j= ∂x′ i∂x ′ i∂x j = L ij,961