is a covariant vector field, then expressions formed by taking
differences of relevant partial derivatives (see 278)
are the components of an antisymmetric covariant tensor field
of order 2, called the
curl of A
. Generally, if there is given an antisymmetric covariant
tensor field of order r, its exterior derivative is a
summation of products of certain of its partial derivatives
and the function sign (see 279)
in which sgn is the sign of the permutation under it and the
summation is over all the j's. The exterior derivative
is best expressed in terms of
exterior differential
forms. Such a form of degree r can be represented by
a linear combination involving products of differentials (see
280) in which the multiplication
of the differentials is antisymmetric (see 281)
and the coefficients are supposed to be antisymmetric in their
indices so that they are completely determined. By the use of
the transformation law of the differentials (see 282)
an exterior differential form of degree r can be
identified with an antisymmetric covariant tensor field of order
r. The exterior derivative can then be written (see
283).
The basic reason for the operation of exterior differentiation
lies in geometry. In fact, exterior differential forms are the
integrands of multiple integrals. The integral of an arbitrary
quantity 
over
an r-dimensional domain D includes as special
cases, for r = 1, n = 2 and r = 2, n
= 3, respectively, the line integral in the plane and the surface
integral in space. If D is the boundary of an (r
+ 1)-dimensional domain F, then the integral of
over domain D is equal to the integral of d
over domain F (see 284),
which is a generalization of the Gauss-Green-Stokes formulas
(named after Gauss and the British mathematicians George Green
and George Gabriel Stokes).
Another important operation for tensor fields is the
Lie derivative, first
stated by Sophus Lie, a Norwegian mathematician. If X is
a contravariant vector field with the components A
in the local coordinate system x
, in which i, k = 1, . . . , n, and f
is a real-valued function, then Xf = 
A
(
/
x)f
is a new real-valued function, the
directional derivative
of f relative to the vector field X. In this way
X can be regarded as a linear differential operator on
functions. If Y is another vector field, it can be shown
that the matrix [X, Y], equal to XY - YX, is
also a linear combination of
/x
and defines
a vector field. This is called the bracket operation of two
vector fields. More generally, a vector field X defines
an infinitesimal transformation 
x
= A
t, i = 1, .
. . n, and the study of the variation of a tensor field
under this infinitesimal transformation leads to a new tensor
field of the same type. The latter is called the Lie derivative
relative to X. The Lie derivative relative to X of
the tensor field B
{sup ij} is determined by the components (see 285).
Generally a tensor field cannot be differentiated to give a
new tensor field because the local coordinates have no geometrical
meaning. A structure on the manifold that makes this possible
is called an
affine connection or simply
a connection. It is given by a set of components 
{sub ik}
in each chart, which, with a change of local coordinates, follow
a transformation rule (see 286).
These components do not, therefore, define tensor fields. But
they can be used to derive from a tensor field of contravariant
order r and covariant order s a new tensor field
of contravariant order r and covariant order s
+ 1, called the covariant derivative of a tensor field. In the
simple case of a tensor field A
its covariant
derivative has a simply expressed form (see 287).
The tensor field is called parallel along a curve x
= x
(t) if a condition is met (see 288).
In the special case of the affine space where the connection
is defined by {sub
ik}
= 0, parallelism of A
means
the constancy of all the components. A
Riemannian manifold has
a uniquely determined connection, the
Levi-Civita connection
(named after Tullio Levi-Civita, an Italian mathematician),
which is characterized by the two properties: (A) the length
of a vector remains unchanged under parallelism; and (B)
{sub ik}
= {sub ki}
.
