Page 53 - Intro to Tensor Calculus
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General Definition
In general a mixed tensor of rank or order (m + n)
T i 1 i 2 ...i m (1.2.53)
j 1 j 2 ...j n
is contravariant of order m and covariant of order n if it obeys the transformation law
x
h i W ∂x i 1 ∂x i 2 ∂x i m ∂x b 1 ∂x b 2 ∂x b n
i 1 i 2 ...i m a 1a 2 ...a m
T = J T ··· · ··· (1.2.54)
x ∂x 1 ∂x 2 ∂x m ∂x ∂x ∂x
j 1 j 2 ...j n b 1 b 2 ...b n a a a j 1 j 2 j n
where
x ∂x ∂(x ,x ,... ,x )
1 2 N
J = = 1 2 N
x ∂x ∂(x , x ,... , x )
is the Jacobian of the transformation. When W = 0 the tensor is called an absolute tensor, otherwise it is
called a relative tensor of weight W.
Here superscripts are used to denote contravariant components and subscripts are used to denote covari-
ant components. Thus, if we are given the tensor components in one coordinate system, then the components
in any other coordinate system are determined by the transformation law of equation (1.2.54). Throughout
the remainder of this text one should treat all tensors as absolute tensors unless specified otherwise.
Dyads and Polyads
Note that vectors can be represented in bold face type with the notation
A = A i E i
This notation can also be generalized to tensor quantities. Higher order tensors can also be denoted by bold
face type. For example the tensor components T ij and B ijk can be represented in terms of the basis vectors
i
E ,i =1,... ,N by using a notation which is similar to that for the representation of vectors. For example,
i
T = T ij E E j
i
j
k
B = B ijk E E E .
Here T denotes a tensor with components T ij and B denotes a tensor with components B ijk . The quantities
j
i
j
i
k
E E are called unit dyads and E E E are called unit triads. There is no multiplication sign between the
basis vectors. This notation is called a polyad notation. A further generalization of this notation is the
representation of an arbitrary tensor using the basis and reciprocal basis vectors in bold type. For example,
a mixed tensor would have the polyadic representation
l
m
n
T = T ij...k E i E j ... E k E E ... E .
lm...n
A dyadic is formed by the outer or direct product of two vectors. For example, the outer product of the
vectors
2
1
2
1
a = a 1 E + a 2 E + a 3 E 3 and b = b 1 E + b 2 E + b 3 E 3