Page 44 - Intro to Tensor Calculus

40

Scalars, Vectors and Tensors

Tensors are quantities which obey certain transformation laws. That is, scalars, vectors, matrices
and higher order arrays can be thought of as components of a tensor quantity. We shall be interested in
finding how these components are represented in various coordinate systems. We desire knowledge of these
transformation laws in order that we can represent various physical laws in a form which is independent of
the coordinate system chosen. Before defining different types of tensors let us examine what we mean by a
coordinate transformation.
Coordinate transformations of the type found in equations (1.2.8) and (1.2.9) can be generalized to
i
higher dimensions. Let x ,i =1, 2,...,N denote N variables. These quantities can be thought of as
2
1
N
representing a variable point (x ,x ,... ,x )in an N dimensional space V N . Another set of N quantities,
1
2
i
N
call them barred quantities, x ,i =1, 2,... ,N, can be used to represent a variable point (x , x ,..., x )in
0
an N dimensional space V N . When the x s are related to the x s by equations of the form
0
N
2
i
1
i
x = x (x , x ,..., x ), i =1, 2,...,N (1.2.30)
i
i
then a transformation is said to exist between the coordinates x and x ,i =1, 2,... ,N. Whenever the
relations (1.2.30) are functionally independent, single valued and possess partial derivatives such that the
Jacobian of the transformation
1 ∂x 1 1
∂x ... ∂x
∂x 1 ∂x 2 N
2
1
x ,x ,...,x N ∂x
x

J = J = . . . . . . (1.2.31)
1
2
x x , x ,..., x N . . ... .
N
∂x N ∂x N ... ∂x
∂x 1 ∂x 2 ∂x N
is different from zero, then there exists an inverse transformation
2
1
N
i
i
x = x (x ,x ,...,x ), i =1, 2,...,N. (1.2.32)
For brevity the transformation equations (1.2.30) and (1.2.32) are sometimes expressed by the notation
i
i
i
i
x = x (x),i =1,... ,N and x = x (x),i =1,... ,N. (1.2.33)
Consider a sequence of transformations from x to ¯x and then from ¯ to ¯ ¯ coordinates. For simplicity
x
x
let ¯x = y and ¯ ¯x = z. If we denote by T 1 ,T 2 and T 3 the transformations
1
N
i
i
T 1 : y = y (x ,...,x ) i =1,... ,N or T 1x = y
1
i
i
N
T 2 : z = z (y ,...,y ) i =1,... ,N or T 2y = z
Then the transformation T 3 obtained by substituting T 1 into T 2 is called the product of two successive
transformations and is written
1
N
N
i
i
N
1
1
T 3 : z = z (y (x ,...,x ),...,y (x ,...,x )) i =1,... ,N or T 3 x = T 2 T 1x = z.
This product transformation is denoted symbolically by T 3 = T 2 T 1 .
The Jacobian of the product transformation is equal to the product of Jacobians associated with the
product transformation and J 3 = J 2 J 1 .