Page 49 - Intro to Tensor Calculus
P. 49
45
i
Letting T ,i =1,...,N denote the components of this tangent vector in the barred system of coordinates,
the equation (1.2.39) can then be expressed in the form
i ∂x i j
T = T , i, j =1,... ,N. (1.2.40)
∂x j
This equation is said to define the transformation law associated with an absolute contravariant tensor of
rank or order one. In the case N = 3 the matrix form of this transformation is represented
1 1 1
1 ∂x ∂x ∂x 1
T ∂x 1 ∂x 2 ∂x 3 T
2 ∂x 2 ∂x 2 ∂x 2
T =
T 2 (1.2.41)
3 ∂x 1 3 ∂x 2 3 ∂x 3 3
3
T ∂x ∂x ∂x T
∂x 1 ∂x 2 ∂x 3
A more general definition is
i
Definition: (Contravariant tensor) Whenever N quantities A in
i
1
N
a coordinate system (x ,... ,x ) are related to N quantities A in a
1
N
coordinate system (x ,... , x ) such that the Jacobian J is different
from zero, then if the transformation law
i W ∂x i j
A = J A
∂x j
is satisfied, these quantities are called the components of a relative tensor
of rank or order one with weight W. Whenever W = 0 these quantities
are called the components of an absolute tensor of rank or order one.
We see that the above transformation law satisfies the group properties.
EXAMPLE 1.2-3. (Transitive Property of Contravariant Transformation)
Show that successive contravariant transformations is also a contravariant transformation.
Solution: Consider the transformation of a vector from an unbarred to a barred system of coordinates. A
i
i
vector or absolute tensor of rank one A = A (x),i =1,... ,N will transform like the equation (1.2.40) and
i ∂x i j
A (x)= A (x). (1.2.42)
∂x j
Another transformation from x → x coordinates will produce the components
i
i ∂x j
A (x)= j A (x) (1.2.43)
∂x
j
j
Here we have used the notation A (x) to emphasize the dependence of the components A upon the x
coordinates. Changing indices and substituting equation (1.2.42) into (1.2.43) we find
i
i ∂x ∂x j
m
A (x)= A (x). (1.2.44)
j
∂x ∂x m