Page 51 - Intro to Tensor Calculus
P. 51
47
Definition: (Covariant tensor) Whenever N quantities A i in a
N
1
coordinate system (x ,...,x ) are related to N quantities A i in a co-
1
N
ordinate system (x ,..., x ), with Jacobian J different from zero, such
that the transformation law
∂x j
W
A i = J i A j (1.2.48)
∂x
is satisfied, then these quantities are called the components of a relative
covariant tensor of rank or order one having a weight of W. When-
ever W = 0, these quantities are called the components of an absolute
covariant tensor of rank or order one.
Again we note that the above transformation satisfies the group properties. Absolute tensors of rank or
order one are referred to as vectors while absolute tensors of rank or order zero are referred to as scalars.
EXAMPLE 1.2-4. (Transitive Property of Covariant Transformation)
Consider a sequence of transformation laws of the type defined by the equation (1.2.47)
∂x j
x → x A i (x)= A j (x) i
∂x
x → x A k (x)= A m (x) ∂x m
∂x k
We can therefore express the transformation of the components associated with the coordinate transformation
x → x and
j m j
∂x ∂x ∂x
A k (x)= A j (x) m k = A j (x) k ,
∂x ∂x ∂x
which demonstrates the transitive property of a covariant transformation.
Higher Order Tensors
We have shown that first order tensors are quantities which obey certain transformation laws. Higher
order tensors are defined in a similar manner and also satisfy the group properties. We assume that we are
given transformations of the type illustrated in equations (1.2.30) and (1.2.32) which are single valued and
i
i
continuous with Jacobian J different from zero. Further, the quantities x and x ,i =1,... ,n represent the
coordinates in any two coordinate systems. The following transformation laws define second order and third
order tensors.