Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.12 PHYSICAL APPLICATIONS OF TENSORS section 26.7, sinceJandωare vectors). The tensor is called theinertia tensoratO of the as ...

TENSORS (iii) referred to these axes as coordinate axes, the inertia tensor is diagonal with diagonal entriesλ 1 ,λ 2 ,λ 3. Two ...

26.12 PHYSICAL APPLICATIONS OF TENSORS We can extend the idea of a second-order tensor that relates two vectors to a situation w ...

TENSORS Further, Poisson’s ratio is defined asσ=−e 22 /e 11 (or−e 33 /e 11 ) and is thus σ= ( 1 e 11 ) λθ 2 μ = ( 1 e 11 )( λ 2 ...

26.14 Non-Cartesian coordinates The other integral theorems discussed in chapter 11 can be extended in a similar way. For exampl ...

TENSORS contrary is specifically stated). All other aspects of the summation convention remain unchanged. With the introduction ...

26.15 The metric tensor second-order tensorT. Using the outer product notation in (26.23), we may write Tin three different ways ...

TENSORS
Calculate the elementsgijof the metric tensor for cylindrical polar coordinates. Hence find the square of the infinites ...

26.15 THE METRIC TENSOR where we have used the reciprocity relation (26.54). Similarly, we could write a·b=aiei·bjej=aibjδji=aib ...

TENSORS Thus, by inverting the matrixGin (26.60), we find that the elementsgijare given in cylindrical polar coordinates by Gˆ=[ ...

26.16 GENERAL COORDINATE TRANSFORMATIONS AND TENSORS so that the basis vectors in the old and new coordinate systems are related ...

TENSORS where the elementsLijare given by L=   cosθ sinθ 0 −sinθ cosθ 0 001  . Thus (26.68) and (26.70) agree with our earli ...

26.17 Relative tensors
Show that the quantitiesgij=ei·ejform the covariant components of a second-order tensor. In the new (pri ...

TENSORS uito anotheru′i, we may define the Jacobian of the transformation (see chapter 6) as the determinant of the transformati ...

26.18 Derivatives of basis vectors and Christoffel symbols the outer product of the two tensors, or any contraction of them, is ...

TENSORS
Using (26.76), deduce the way in which the quantitiesΓkijtransform under a general coordinate transformation, and hence ...

26.18 DERIVATIVES OF BASIS VECTORS AND CHRISTOFFEL SYMBOLS where we have used the definition (26.75). By cyclically permuting th ...

TENSORS 26.19 Covariant differentiation For Cartesian tensors we noted that the derivative of a scalar is a (covariant) vector.T ...

26.19 COVARIANT DIFFERENTIATION constant (this term vanishes in Cartesian coordinates). Using (26.75) we write ∂v ∂uj = ∂vi ∂uj ...

TENSORS and so vi;i= ∂vρ ∂ρ + ∂vφ ∂φ + ∂vz ∂z + 1 ρ vρ = 1 ρ ∂ ∂ρ (ρvρ)+ ∂vφ ∂φ + ∂vz ∂z . This result is identical to the expre ...