Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.3 CARTESIAN TENSORS O x 1 x 2 x′ 1 x′ 2 θ θ θ Figure 26.1 Rotation of Cartesian axes by an angleθabout thex 3 -axis. The thre ...

TENSORS 26.4 First- and zero-order Cartesian tensors Using the above example as a guide, we may consider any set of three quanti ...

26.4 FIRST- AND ZERO-ORDER CARTESIAN TENSORS (ii) Herev 1 =x 2 andv 2 =x 1. Following the same procedure, v 1 ′=x′ 2 =−sx 1 +cx ...

TENSORS In fact any scalar product of two first-order tensors (vectors) is a zero-order tensor (scalar), as might be expected si ...

26.5 Second- and higher-order Cartesian tensors
Ifviare the components of a first-order tensor, show that∇·v=∂vi/∂xiis a zero-o ...

TENSORS another, without reference to any coordinate system) and consider the matrix containing its components as a representati ...

26.5 SECOND- AND HIGHER-ORDER CARTESIAN TENSORS (ii)The gradient of a vector.Supposevirepresents the components of a vector; let ...

TENSORS Physical examples involving second-order tensors will be discussed in the later sections of this chapter, but we might n ...

26.7 The quotient law An operation that produces the opposite effect – namely, generates a tensor of smaller rather than larger ...

TENSORS does this imply that theApq···k···malso form the components of a tensorA?Here A,BandCare respectively ofMth,Nth and (M+N ...

26.8 The tensorsδijandijk Nsubscripts, with an arbitraryNth-order tensor (i.e. one having independently variable components) an ...

TENSORS Let us begin, however, by noting that we may use the Levi–Civita symbol to write an expression for the determinant of a ...

26.8 THE TENSORSδijANDijk
Write the following as contracted Cartesian tensors:a·b,∇^2 φ,∇×v,∇(∇·v),∇×(∇×v), (a×b)·c. The corre ...

TENSORS A useful application of (26.30) is in obtaining alternative expressions for vector quantities that arise from the vector ...

26.9 ISOTROPIC TENSORS are independent of the transformationLij. Specifically,δ 11 has the value 1 in all coordinate frames, whe ...

TENSORS Rotate byπ/2 about theOx 3 -axis:L 12 =−1,L 21 =1,L 33 = 1, the otherLij=0. (d)T 111 =(−1)×(−1)×(−1)×T 222 =−T 222 , (e) ...

26.10 IMPROPER ROTATIONS AND PSEUDOTENSORS O O p v x 1 x 2 x 3 x′ 1 x′ 2 x′ 3 v′ p′ Figure 26.2 The behaviour of a vectorvand a ...

TENSORS but since|L|=±1 we may rewrite this as ijk=|L|LilLjmLknlmn. From this expression, we see that althoughijkbehaves as a ...

26.11 Dual tensors formations, for which the physical system of interest is left unaltered, and only the coordinate system used ...

TENSORS
Using (26.40), show thatAij=ijkpk. By contracting both sides of (26.40) withijk, we find ijkpk=^12 ijkklmAlm. Usin ...