Vector operators in tensor form

26.20 VECTOR OPERATORS IN TENSOR FORM
where the expression in parentheses is the required covariant derivative
T ij ; k =
∂T ij
+ Γilk T lj + Γ j lk T il .
∂uk
(26.89)
k
Using (26.89), the derivative of the tensor T with respect to u can now be written in terms
of its contravariant components as
∂T
= T ij ; k ei ⊗ ej . ∂uk
Results similar to (26.89) may be obtained for the the covariant derivatives of
the mixed and covariant components of a second-order tensor. Collecting these
results together, we have
T ij ; k = T ij , k + Γilk T lj + Γ j lk T il ,
T ij; k = T ij, k + Γilk T l j − Γl jk T il ,
Tij; k = Tij, k − Γl ik Tlj − Γl jk Til ,
where we have used the comma notation for partial derivatives. The position of
the indices in these expressions is very systematic: for each contravariant index
(superscript) on the LHS we add a term on the RHS containing a Christoffel
symbol with a plus sign, and for every covariant index (subscript) we add a
corresponding term with a minus sign. This is extended straightforwardly to
tensors with an arbitrary number of contravariant and covariant indices.
We note that the quantities T ij ; k , T ij; k and Tij; k are the components of the
same third-order tensor ∇T with respect to different tensor bases, i.e.
∇T = T ij ; k ei ⊗ ej ⊗ ek = T ij; k ei ⊗ e j ⊗ ek = Tij; k ei ⊗ e j ⊗ ek .
We conclude this section by considering briefly the covariant derivative of a
scalar. The covariant derivative differs from the simple partial derivative with
respect to the coordinates only because the basis vectors of the coordinate
system change with position in space (hence for Cartesian coordinates there is no
difference). However, a scalar φ does not depend on the basis vectors at all and
so its covariant derivative must be the same as its partial derivative, i.e.
φ; j =
∂φ
= φ, j .
∂u j
(26.90)
26.20 Vector operators in tensor form
In section 10.10 we used vector calculus methods to find expressions for vector
differential operators, such as grad, div, curl and the Laplacian, in general orthogonal curvilinear coordinates, taking cylindrical and spherical polars as particular
examples. In this section we use the framework of general tensors that we have
developed to obtain, in tensor form, expressions for these operators that are valid
in all coordinate systems, whether orthogonal or not.
971
TENSORS
In order to compare the results obtained here with those given in section
10.10 for orthogonal coordinates, it is necessary to remember that here we are
working with the (in general) non-unit basis vectors ei = ∂r/∂ui or ei = ∇ui .
Thus the components of a vector v = v i ei are not the same as the components v̂ i
appropriate to the corresponding unit basis êi . In fact, if the scale factors of the
coordinate system are hi , i = 1, 2, 3, then v i = v̂ i /hi (no summation over i).
As mentioned in section 26.15, for an orthogonal coordinate system with scale
factors hi we have
#
#
h2i if i = j,
1/h2i if i = j,
and
g ij =
gij =
0 otherwise
0
otherwise,
and so the determinant g of the matrix [gij ] is given by g = h21 h22 h23 .
Gradient
The gradient of a scalar φ is given by
∇φ = φ; i ei =
∂φ i
e,
∂ui
(26.91)
since the covariant derivative of a scalar is the same as its partial derivative.
Divergence
Replacing the partial derivatives that occur in Cartesian coordinates with covariant derivatives, the divergence of a vector field v in a general coordinate system
is given by
∇ · v = vi ; i =
∂v i
+ Γiki v k .
∂ui
Using the expression (26.82) for the Christoffel symbol in terms of the metric
tensor, we find
∂gil
∂gil
∂gkl
∂gki
+
−
(26.92)
Γiki = 12 g il
= 12 g il k .
∂uk
∂ui
∂ul
∂u
The last two terms have cancelled because
g il
∂gkl
∂gki
∂gki
= g li l = g il l ,
∂ui
∂u
∂u
where in the first equality we have interchanged the dummy indices i and l, and
in the second equality have used the symmetry of the metric tensor.
We may simplify (26.92) still further by using a result concerning the derivative
of the determinant of a matrix whose elements are functions of the coordinates.
972
26.20 VECTOR OPERATORS IN TENSOR FORM
Suppose A = [aij ], B = [bij ] and that B = A−1 . By considering the determinant a = |A|,
show that
∂aij
∂a
= ab ji k .
∂uk
∂u
If we denote the cofactor of the element aij by ∆ij then the elements of the inverse matrix
are given by (see chapter 8)
bij =
1 ji
∆ .
a
(26.93)
However, the determinant of A is given by
aij ∆ij ,
a=
j
in which we have fixed i and written the sum over j explicitly, for clarity. Partially
differentiating both sides with respect to aij , we then obtain
∂a
= ∆ij ,
∂aij
(26.94)
since aij does not occur in any of the cofactors ∆ij .
Now, if the aij depend on the coordinates then so will the determinant a and, by the
chain rule, we have
∂a ∂aij
∂a
∂aij
∂aij
=
= ∆ij k = ab ji k ,
∂uk
∂aij ∂uk
∂u
∂u
(26.95)
in which we have used (26.93) and (26.94). Applying the result (26.95) to the determinant g of the metric tensor, and
remembering both that g ik gkj = δji and that g ij is symmetric, we obtain
∂gij
∂g
= gg ij k .
(26.96)
∂uk
∂u
Substituting (26.96) into (26.92) we find that the expression for the Christoffel
symbol can be much simplified to give
√
1 ∂g
1 ∂ g
=
.
Γiki =
√
2g ∂uk
g ∂uk
Thus finally we obtain the expression for the divergence of a vector field in a
general coordinate system as
1 ∂ √ j
( gv ).
∇ · v = vi ; i = √
g ∂u j
(26.97)
Laplacian
If we replace v by ∇φ in ∇ · v then we obtain the Laplacian ∇2 φ. From (26.91),
we have
∂φ
vi ei = v = ∇φ = i ei ,
∂u
973
TENSORS
and so the covariant components of v are given by vi = ∂φ/∂ui . In (26.97),
however, we require the contravariant components v i . These may be obtained by
raising the index using the metric tensor, to give
v j = g jk vk = g jk
Substituting this into (26.97) we obtain
1 ∂
∇2 φ = √
g ∂u j
√
∂φ
.
∂uk
gg jk
∂φ
∂uk
.
(26.98)
Use (26.98) to find the expression for ∇2 φ in an orthogonal coordinate system with scale
factors hi , i = 1, 2, 3.
√
For an orthogonal coordinate system g = h1 h2 h3 ; further, g ij = 1/h2i if i = j and g ij = 0
otherwise. Therefore, from (26.98) we have
1
h1 h2 h3 ∂φ
∂
,
∇2 φ =
h1 h2 h3 ∂u j
h2j ∂u j
which agrees with the results of section 10.10. Curl
The special vector form of the curl of a vector field exists only in three dimensions.
We therefore consider a more general form valid in higher-dimensional spaces as
well. In a general space the operation curl v is defined by
(curl v)ij = vi; j − vj; i ,
which is an antisymmetric covariant tensor.
In fact the difference of derivatives can be simplified, since
∂vi
∂vj
− Γl ij vl − i + Γl ji vl
∂u j
∂u
∂vi
∂vj
=
− i,
∂u j
∂u
where the Christoffel symbols have cancelled because of their symmetry properties.
Thus curl v can be written in terms of partial derivatives as
vi; j − vj; i =
∂vi
∂vj
− i.
∂u j
∂u
Generalising slightly the discussion of section 26.17, in three dimensions we may
associate with this antisymmetric second-order tensor a vector with contravariant
components,
(curl v)ij =
1
(∇ × v)i = − √ ijk (curl v)jk
2 g
∂vj
∂vk
1
∂vk
1
= − √ ijk
−
= √ ijk j ;
2 g
∂uk
∂u j
g
∂u
974