6.1 Covariant Derivatives, Component View
83
As usual we have made liberal use of relabeling summation indices. Finally we
combine the last two equations to obtain the complete transformation, written in
three equivalent ways,
∂ W
i
∂ x
k
+
i
k j W
j =
∂ x
i
∂ x l
∂ x
j
∂ x
k
∂ W
l
∂ x j +
l
jn W
n
,
(6.6a)
W
i
,k +
i
k j W
j =
∂ x
i
∂ x l
∂ x
j
∂ x
k
W
l
, j +
l
jn W
n
,
(6.6b)
W
i
;k =
∂ x
i
∂ x l
∂ x
j
∂ x
k
W
l
; j .
(6.6c)
The second derivative term has magically cancelled out. The transformation law is
that of a second rank tensor, once contravariant and once covariant or (1,1).
This theorem makes it clear that the covariant derivative is the natural generalization of the ordinary derivative since it is a tensor and reduces to the ordinary
derivative in flat space with Cartesian coordinates.
We now have derivatives for a scalar field and for a vector field that are tensors.
From these we can infer unique definitions for derivatives of other tensors. To obtain
a definition for the covariant derivative of a covariant vector field we use what we
already know about the derivatives of the scalar and the contravariant vector fields;
we demand that the product rule (or Leibniz rule) for ordinary derivatives hold also
for the covariant derivative. Thus both the ordinary and covariant derivative of the
scalar field W
k V k should obey the product rule. This gives
(W
k V k ) ,l = W
k ,l V k + W
k V k,l ordinary derivative,
(W
k V k ) ;l = W
k ;l V k + W
k V k;l covariant derivative.
(6.7)
But for the scalar inner product the ordinary and covariant derivatives are the same,
so
W
k V k;l = W
k V k,l + W
k ,l V k − W
k ;l V k = W
k
V k,l −
n
kl V n
.
(6.8)
Since W
k can be any vector we see that for consistency the covariant derivative must
be defined as
V k;l = V k,l −
n
kl V n .
(6.9)
We have now obtained consistent definitions for the covariant derivative of scalar
fields and contravariant and covariant vector fields. Using these and the product rule
we may infer definitions and properties for the covariant derivative of any (M, N)
tensor field. For example the covariant derivative of a second rank tensor must be the
same as that for the direct product of vectors, both for consistency and because any
83
As usual we have made liberal use of relabeling summation indices. Finally we
combine the last two equations to obtain the complete transformation, written in
three equivalent ways,
∂ W
i
∂ x
k
+
i
k j W
j =
∂ x
i
∂ x l
∂ x
j
∂ x
k
∂ W
l
∂ x j +
l
jn W
n
,
(6.6a)
W
i
,k +
i
k j W
j =
∂ x
i
∂ x l
∂ x
j
∂ x
k
W
l
, j +
l
jn W
n
,
(6.6b)
W
i
;k =
∂ x
i
∂ x l
∂ x
j
∂ x
k
W
l
; j .
(6.6c)
The second derivative term has magically cancelled out. The transformation law is
that of a second rank tensor, once contravariant and once covariant or (1,1).
This theorem makes it clear that the covariant derivative is the natural generalization of the ordinary derivative since it is a tensor and reduces to the ordinary
derivative in flat space with Cartesian coordinates.
We now have derivatives for a scalar field and for a vector field that are tensors.
From these we can infer unique definitions for derivatives of other tensors. To obtain
a definition for the covariant derivative of a covariant vector field we use what we
already know about the derivatives of the scalar and the contravariant vector fields;
we demand that the product rule (or Leibniz rule) for ordinary derivatives hold also
for the covariant derivative. Thus both the ordinary and covariant derivative of the
scalar field W
k V k should obey the product rule. This gives
(W
k V k ) ,l = W
k ,l V k + W
k V k,l ordinary derivative,
(W
k V k ) ;l = W
k ;l V k + W
k V k;l covariant derivative.
(6.7)
But for the scalar inner product the ordinary and covariant derivatives are the same,
so
W
k V k;l = W
k V k,l + W
k ,l V k − W
k ;l V k = W
k
V k,l −
n
kl V n
.
(6.8)
Since W
k can be any vector we see that for consistency the covariant derivative must
be defined as
V k;l = V k,l −
n
kl V n .
(6.9)
We have now obtained consistent definitions for the covariant derivative of scalar
fields and contravariant and covariant vector fields. Using these and the product rule
we may infer definitions and properties for the covariant derivative of any (M, N)
tensor field. For example the covariant derivative of a second rank tensor must be the
same as that for the direct product of vectors, both for consistency and because any