Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSu i to another u ′i , we may define the Jacobian of the transformation (see chapter 6)as the determinant of the transformation matrix [∂u ′i /∂u j ]: this is usually denotedbyJ =∂u ′∣ ∂u ∣ .Alternatively, we may interchange the primed and unprimed coordinates toobtain |∂u/∂u ′ | =1/J: unfortunately this also is often called the Jacobian of thetransformation.Using the Jacobian J, we define a relative tensor of weight w as one whosecomponents transform as follows:T ′ ij···klm···n = ∂u′ i∂u ′ j···∂u′ k∣∂u d ∂u e ∂uf∂u a ∂u b ∂u c ∂u ′ l∂u ′ m ···∂u ′ n T ab···c ∂u ∣∣∣wde···f ∣∂u ′ .(26.74)Comparing this expression with (26.71), we see that a true (or absolute) generaltensor may be considered as a relative tensor of weight w =0.Ifw = −1, on theother hand, the relative tensor is known as a general pseudotensor and if w =1as a tensor density.It is worth comparing (26.74) with the definition (26.39) of a Cartesian pseudotensor.For the latter, we are concerned only with its behaviour under a rotation(proper or improper) of Cartesian axes, for which the Jacobian J = ±1. Thus,general relative tensors of weight w = −1 andw = 1 would both satisfy thedefinition (26.39) of a Cartesian pseudotensor.◮If the g ij are the covariant components of the metric tensor, show that the determinant gof the matrix [g ij ] is a relative scalar of weight w =2.The components g ij transform asg ij ′ = ∂uk ∂u l∂u ′i ∂u g kl.′jDefining the matrices U =[∂u i /∂u ′j ], G =[g ij ]andG ′ =[g ij ′ ], we may write this expressionasG ′ = U T GU.Taking the determinant of both sides, we obtain∣ g ′ = |U| 2 g =∂u ∣∣∣2∣ g,∂u ′which shows that g is a relative scalar of weight w =2.◭From the discussion in section 26.8, it can be seen that ɛ ijk is a covariantrelative tensor of weight −1. We may also define the contravariant tensor ɛ ijk ,which is numerically equal to ɛ ijk but is a relative tensor of weight +1.If two relative tensors have weights w 1 and w 2 respectively then, from (26.74),964