Page 52 - Intro to Tensor Calculus

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Definition: (Second order contravariant tensor) Whenever N-squared quantities A ij
mn
1
N
in a coordinate system (x ,...,x ) are related to N-squared quantities A in a coordinate
1
N
system (x ,... , x ) such that the transformation law
mn ij W ∂x m ∂x n
A (x)= A (x)J (1.2.49)
i
∂x ∂x j
is satisfied, then these quantities are called components of a relative contravariant tensor of
rank or order two with weight W. Whenever W = 0 these quantities are called the components
of an absolute contravariant tensor of rank or order two.

Definition: (Second order covariant tensor) Whenever N-squared quantities
1 N
A ij in a coordinate system (x ,... ,x ) are related to N-squared quantities A mn
N
1
in a coordinate system (x ,..., x ) such that the transformation law
i
∂x ∂x j
W
A mn (x)= A ij (x)J m n (1.2.50)
∂x ∂x
is satisfied, then these quantities are called components of a relative covariant tensor
of rank or order two with weight W. Whenever W = 0 these quantities are called
the components of an absolute covariant tensor of rank or order two.

Definition: (Second order mixed tensor) Whenever N-squared quantities
1
N
i
A in a coordinate system (x ,... ,x ) are related to N-squared quantities A m in
j n
1
N
a coordinate system (x ,... , x ) such that the transformation law
m i W ∂x m ∂x j
A (x)= A (x)J n (1.2.51)
n
j
i
∂x ∂x
is satisfied, then these quantities are called components of a relative mixed tensor of
rank or order two with weight W. Whenever W = 0 these quantities are called the
components of an absolute mixed tensor of rank or order two. It is contravariant
of order one and covariant of order one.

Higher order tensors are defined in a similar manner. For example, if we can find N-cubed quantities
A m such that
np
i
α
i γ W ∂x ∂x ∂x β
A (x)= A (x)J (1.2.52)
jk αβ γ j k
∂x ∂x ∂x
then this is a relative mixed tensor of order three with weight W. It is contravariant of order one and
covariant of order two.