Page 104 - Intro to Tensor Calculus

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(See exercise 1.1, problem 15) the trilinear form, given by equation (1.3.71), with vectors from equations
(1.3.72), can be expressed as
i j k
ϕ(~x, ~y,~)= e ijk x y z , i,j,k =1, 2, 3. (1.3.75)
z
The coefficients e ijk of the trilinear form is called a third order tensor. It is the familiar permutation symbol
considered earlier.
z
In a multilinear form of degree M, ϕ(~x, ~y,... ,~), the M arguments can be represented in a component
i
i
i
form with respect to a set of basis vectors ( b e 1 , b e 2 , b e 3 ). Let these vectors have components x ,y ,z ,i =1, 2, 3
with respect to the selected basis vectors. We then can write
j
k
i
~x = x b e i , ~y = y b e j , ~ = z b e k .
z
Substituting these vectors into the M degree multilinear form produces
i
i j
k
j
k
ϕ(x b e i ,y b e j ,... ,z b e k )= x y ··· z ϕ( b e i , b e j ,..., b e k ). (1.3.76)
Consequently, the multilinear form defines a set of coefficients
a ij...k = ϕ( b e i , b e j ,..., b e k ) (1.3.77)
which are referred to as the components of a tensor of order M. The tensor is thus created by the multilinear
form and has M indices if ϕ is of degree M.
~
~
~
Note that if we change to a different set of basis vectors, say, (E 1 , E 2 , E 3 ) the multilinear form defines
anew tensor
~
~
~
a ij...k = ϕ(E i , E j ,... , E k ). (1.3.78)
This new tensor has a bar over it to distinguish it from the previous tensor. A definite relation exists between
the new and old basis vectors and consequently there exists a definite relation between the components of
the barred and unbarred tensors components. Recall that if we are given a set of transformation equations
1
i
3
i
2
y = y (x ,x ,x ),i =1, 2, 3, (1.3.79)
from rectangular to generalized curvilinear coordinates, we can express the basis vectors in the new system
by the equations
∂y j
~
E i = b e j , i =1, 2, 3. (1.3.80)
∂x i
3
3
2
1
2
1
For example, see equations (1.3.11) with y = x, y = y, y = z, x = u, x = v, x = w. Substituting
equations (1.3.80) into equations (1.3.78) we obtain
∂y α ∂y β ∂y γ
a ij...k = ϕ( b e α , b e β ,... , b e γ ).
∂x i ∂x j ∂x k
By the linearity property of ϕ, this equation is expressible in the form
α
∂y ∂y β ∂y γ
a ij...k = ... ϕ( b e α , b e β ,..., b e γ )
i
∂x ∂x j ∂x k
α
∂y ∂y β ∂y γ
a ij...k = ... a αβ...γ
i
∂x ∂x j ∂x k
This is the familiar transformation law for a covariant tensor of degree M. By selecting reciprocal basis
vectors the corresponding transformation laws for contravariant vectors can be determined.
The above examples illustrate that tensors can be considered as quantities derivable from multilinear
forms defined on some vector space.