Page 98 - Intro to Tensor Calculus

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Higher Order Tensors

The physical components associated with higher ordered tensors are defined by projections in V N just
i
i
like the case with first order tensors. For an nth ordered tensor T ij...k we can select n unit vectors λ ,µ ,... ,ν i
and form the inner product (projection)
i j
k
T ij...k λ µ ...ν .
When projecting the tensor components onto the coordinate curves, there are N choices for each of the unit
n
vectors. This produces N physical components.
The above inner product represents the physical component of the tensor T ij...k along the directions of
i
i
i
the unit vectors λ ,µ ,... ,ν . The selected unit vectors may or may not be orthogonal. In the cases where
the selected unit vectors are all orthogonal to one another, the calculation of the physical components is
greatly simplified. By relabeling the unit vectors λ i ,λ i ,... ,λ i where (m), (n), ..., (p) represent one of
(m) (n) (p)
the N directions, the physical components of a general nth order tensor is represented

T (mn... p)= T ij...k λ i λ j ...λ k
(m) (n) (p)
EXAMPLE 1.3-12. (Physical components)
In an orthogonal curvilinear coordinate system V 3 with metric g ij ,i, j =1, 2, 3, find the physical com-
ponents of
i
ij
(i) the second order tensor A ij . (ii) the second order tensor A . (iii) the second order tensor A .
j
i
Solution: The physical components of A mn ,m,n =1, 2, 3 along the directions of two unit vectors λ and
i
µ is defined as the inner product in V 3 . These physical components can be expressed
m
A(ij)= A mn λ µ n i, j =1, 2, 3,
(i) (j)
where the subscripts (i)and (j) represent one of the coordinate directions. Dropping the subscripts (i)and
(j), we make the observation that in an orthogonal curvilinear coordinate system there are three choices for
i
i
the direction of the unit vector λ and also three choices for the direction of the unit vector µ . These three
1
2
3
choices represent the directions along the x ,x or x coordinate curves which emanate from a point of the
curvilinear coordinate system. This produces a total of nine possible physical components associated with
the tensor A mn .
1
i
For example, we can obtain the components of the unit vector λ ,i =1, 2, 3in the x direction directly
from an examination of the element of arc length squared
2
3 2
1 2
2
2
2
2 2
ds =(h 1 ) (dx ) +(h 2 ) (dx ) +(h 3 ) (dx ) .
3
2
By setting dx = dx = 0, we find
dx 1 1 1 2 3
= = λ , λ =0, λ =0.
ds h 1
i
This is the vector λ i ,i =1, 2, 3. Similarly, if we choose to select the unit vector λ ,i =1, 2, 3inthe x 2
(1)
1
3
direction, we set dx = dx = 0 in the element of arc length squared and find the components
dx 2 1
1 2 3
λ =0, λ = = , λ =0.
ds h 2