A branch of
vector calculus
in which scalar and vector fields are studied (cf.
Scalar field;
Vector field).
One of the fundamental concepts in vector analysis for the study of scalar fields is the
gradient.
A scalar field
is said to be differentiable at a point
of a domain
if the increment of the field,
,
at
may be written as
where
is the vector connecting the points
and
,
is the distance between
and
and
is a linear form applied to the vector
.
The linear form
may be uniquely represented as
where
is a vector which does not depend on
(i.e. on the choice of
).
The vector
is said to be the
gradient
of the scalar field and is denoted by the symbol
.
If the scalar field is differentiable at every point of some domain,
is a vector field. The direction of the gradient
is always orthogonal to the level lines (surfaces)
of the scalar field
,
with the directional derivative given by
The concepts of
divergence
and
curl
are also employed in the study of vector fields. Let a vector field
be differentiable at a point
of a certain domain
,
i.e. the field increment at the point
can be uniquely represented as
where
and
is a linear operator which is independent of
(of the choice of
).
The
divergence
of the vector field
is the following scalar invariant of the linear operator
:
where
are
dual bases:
(
is the Kronecker symbol). If
is the velocity field of a stationary flow of a non-compressible liquid,
at the point
denotes the intensity of the source
(
)
or of the sink
(
)
present at
,
or their absence
(
).
The
curl
(rotor)
of the vector field
on a domain in
is the following vector invariant of the linear operator
from
(*):
where
are dual bases. The curl of a vector field may
be interpreted as the
"rotational component"
of this field.
For vector and scalar fields of class
repeated operations are possible, for example:
where
is the
Laplace operator.
Gradient, divergence and curl together are usually known as the
basic differential operations
of vector analysis. See
Curl;
Gradient;
Divergence
for their properties and expressions in special coordinate systems.
Fundamental integral formulas, connecting volume, surface and contour integrals,
can be written down in terms of the basic
operations of vector analysis. Let a vector field
be continuously differentiable in a bounded connected domain
with piecewise-smooth boundary
.
Let
be a bounded, complete, piecewise-smooth, two-sided
(oriented) surface with piecewise-smooth boundary
.
Then the
Stokes formula
will be applicable:
where the vector
normal to
and the vector
tangent to
must be determined in accordance with the orientations of the surface
and its boundary
.
The integral
is known as the
circulation
of
along
.
If the circulation of a vector field along an arbitrary closed piecewise-smooth curve
in a given domain is zero, the vector field is said to be
potential
(or
conservative)
in this domain. In a simply-connected domain a vector field is conservative if
.
For a conservative vector field there exists the so-called
scalar potential,
which is a function
such that
;
here
where the points
,
is a piecewise-smooth curve in
,
is the unit vector tangent to
,
and
is the line element of
.
Let the vector field
be continuously differentiable in a bounded connected domain
with piecewise-smooth boundary
;
the
Ostrogradski formula
reads as follows:
where
is the exterior normal vector to
.
The integral
is said to be the
flux
of
across
.
If the flux of a vector field across
an arbitrary, piecewise-smooth, non-self-intersecting, oriented surface in
which is the boundary of some bounded subdomain of
is zero, the vector field
is said to be
solenoidal
in
.
For a continuously-differentiable vector field to be
solenoidal it is necessary and sufficient that
at all points of
.
For a solenoidal vector field
there exists a so-called
vector potential:
a function
such that
If the divergence and the curl of a vector field are defined at each point
of a simply-connected domain
,
the vector field can be represented everywhere in
as the sum of a potential field
and a solenoidal field
(Helmholtz' theorem):
Vector fields for which
and
are called
harmonic.
The potential
of a harmonic vector field satisfies the
Laplace equation.
The scalar field
is also said to be
harmonic.
For references, see
Vector calculus.