Page 100 - Intro to Tensor Calculus

95

We use the fact g ij =0 for i 6= j together with the physical components from equation (1.3.63) to reduce the
(i) (i)(m)
above equation to the form A = A no summation on m . In terms of physical components
(m) g (m)(m)
we have
h (m) (i)(m) 2 (i)(m)
A(im)= A h (m) or A(im)= A h (i) h (m) . no summation i, m =1, 2, 3 (1.3.64)
h (i)

Examining the results from equation (1.3.64) we find that the physical components associated with the
ij
contravariant tensor A , in an orthogonal coordinate system, can be written as:
11 12 13
A(11) = A h 1 h 1 A(12) = A h 1 h 2 A(13) = A h 1 h 3
21 22 23
A(21) = A h 2 h 1 A(22) = A h 2 h 2 A(23) = A h 2 h 3
31 32 33
A(31) = A h 3 h 1 A(32) = A h 3 h 2 A(33) = A h 3 h 3 .

Physical Components in General

In an orthogonal curvilinear coordinate system, the physical components associated with the nth order
tensor T ij...kl along the curvilinear coordinate directions can be represented:

T (i)(j)...(k)(l)
T (ij ...kl)= no summations.
h (i) h (j) ... h (k) h (l)

These physical components can be related to the various tensors associated with T ij...kl . For example, in
an orthogonal coordinate system, the physical components associated with the mixed tensor T ij...m can be
n...kl
expressed as:
(i)(j)...(m) h (i) h (j) ...h (m)
T (ij ...mn... kl)= T no summations. (1.3.65)
(n)...(k)(l)
h (n) ...h (k) h (l)
i
i
EXAMPLE 1.3-13. (Physical components) Let x = x (t),i =1, 2, 3 denote the position vector of a
i
i
particle which moves as a function of time t. Assume there exists a coordinate transformation x = x (x), for
i =1, 2, 3, of the form given by equations (1.2.33). The position of the particle when referenced with respect
to the barred system of coordinates can be found by substitution. The generalized velocity of the particle
in the unbarred system is a vector with components

dx i
i
v = ,i =1, 2, 3.
dt
The generalized velocity components of the same particle in the barred system is obtained from the chain
rule. We find this velocity is represented by

i
dx i ∂x dx j ∂x i j
i
v = = = v .
dt ∂x dt ∂x j
j
This equation implies that the contravariant quantities
dx 1 dx 2 dx 3
3
2
1
(v ,v ,v )= ( , , )
dt dt dt