Page 50 - Intro to Tensor Calculus

46

From the fact that
i j i
∂x ∂x ∂x
= ,
j
∂x ∂x m ∂x m
the equation (1.2.44) simplifies to
i
i ∂x
m
A (x)= A (x) (1.2.45)
∂x m
and hence this transformation is also contravariant. We express this by saying that the above are transitive
with respect to the group of coordinate transformations.
Note that from the chain rule one can write

∂x m ∂x j ∂x m ∂x 1 ∂x m ∂x 2 ∂x m ∂x 3 ∂x m m
= + + = = δ .
n
3
2
1
j
∂x ∂x n ∂x ∂x n ∂x ∂x n ∂x ∂x n ∂x n
Do not make the mistake of writing
∂x m ∂x 2 = ∂x m or ∂x m ∂x 3 = ∂x m
2
3
∂x ∂x n ∂x n ∂x ∂x n ∂x n
as these expressions are incorrect. Note that there are no summations in these terms, whereas there is a
summation index in the representation of the chain rule.

Vector Transformation, Covariant Components

Consider a scalar invariant A(x)= A(x) which is a shorthand notation for the equation

1
2
1
2
n
n
A(x ,x ,...,x )= A(x , x ,..., x )
involving the coordinate transformation of equation (1.2.30). By the chain rule we differentiate this invariant
and find that the components of the gradient must satisfy
∂A ∂A ∂x j
= . (1.2.46)
j
∂x i ∂x ∂x i
Let
∂A ∂A
A j = and A i = ,
∂x j ∂x i
then equation (1.2.46) can be expressed as the transformation law
∂x j
A i = A j i . (1.2.47)
∂x
This is the transformation law for an absolute covariant tensor of rank or order one. A more general definition
is