Vector analysis

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 $ u( M) $ is said to be differentiable at a point $ M $ of a domain $ D $ if the increment of the field, $ \Delta u $, at $ M $ may be written as

$$ \Delta u = f ( \Delta {\mathbf r } ) + o ( \rho ) , $$

where $ \mathbf r $ is the vector connecting the points $ M $ and $ M ^ \prime $, $ \rho = \rho ( M, M ^ \prime ) $ is the distance between $ M $ and $ M ^ \prime $ and $ f( \Delta \mathbf r ) $ is a linear form applied to the vector $ \Delta \mathbf r $. The linear form $ f( \Delta \mathbf r ) $ may be uniquely represented as

$$ f ( \Delta \mathbf r ) = ( \mathbf g , \Delta \mathbf r ), $$

where $ \mathbf g $ is a vector which does not depend on $ \Delta \mathbf r $( i.e. on the choice of $ M ^ \prime $). The vector $ \mathbf g $ is said to be the gradient of the scalar field and is denoted by the symbol $ \mathop{\rm grad} u $. If the scalar field is differentiable at every point of some domain, $ \mathop{\rm grad} u $ is a vector field. The direction of the gradient is always orthogonal to the level lines (surfaces) $ u( M) = \textrm{ const } $ of the scalar field $ u $, with the directional derivative given by

$$ \nabla _ {e} u = ( \mathop{\rm grad} u , \mathbf e ). $$

The concepts of divergence and curl are also employed in the study of vector fields. Let a vector field $ \mathbf a ( M) $ be differentiable at a point $ M $ of a certain domain $ D $, i.e. the field increment at the point $ M $ can be uniquely represented as

$$ \Delta {\mathbf a } = A \Delta {\mathbf r } + o ( | \Delta {\mathbf r } | ), $$

where $ \Delta {\mathbf r } = | M M ^ \prime | $ and $ A $ is a linear operator which is independent of $ \Delta \mathbf r $( of the choice of $ M ^ \prime $). The divergence $ \mathop{\rm div} \mathbf a $ of the vector field $ \mathbf a ( M) $ is the following scalar invariant of the linear operator $ A $:

$$ \tag{* } \mathop{\rm div} \mathbf a \equiv ( \mathbf r ^ {i} , A \mathbf r _ {i} ), $$

where $ \mathbf r ^ {i} , \mathbf r _ {i} $ are dual bases: $ ( \mathbf r _ {i} , \mathbf r ^ {k} ) = \delta _ {i} ^ {k} $( $ \delta _ {i} ^ {k} $ is the Kronecker symbol). If $ \mathbf a ( M) $ is the velocity field of a stationary flow of a non-compressible liquid, $ \mathop{\rm div} \mathbf a $ at the point $ M $ denotes the intensity of the source ( $ \mathop{\rm div} \mathbf a > 0 $) or of the sink ( $ \mathop{\rm div} \mathbf a < 0 $) present at $ M $, or their absence ( $ \mathop{\rm div} \mathbf a = 0 $).

The curl (rotor) $ \mathop{\rm rot} \mathbf a $ of the vector field $ \mathbf a ( M) $ on a domain in $ \mathbf R ^ {3} $ is the following vector invariant of the linear operator $ A $ from (*):

$$ \mathop{\rm rot} \mathbf a \equiv [ \mathbf r _ {i} , A \mathbf r ^ {i} ], $$

where $ \mathbf r ^ {i} , \mathbf r _ {i} $ 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 $ C ^ {2} $ repeated operations are possible, for example:

$$ \mathop{\rm rot} \mathop{\rm grad} u = 0, $$

$$ \mathop{\rm div} \mathop{\rm rot} \mathbf a = 0, $$

$$ \mathop{\rm rot} \mathop{\rm rot} \mathbf a = \mathop{\rm grad} \mathop{\rm div} \mathbf a - \Delta {\mathbf a } , $$

$$ \mathop{\rm div} \mathop{\rm grad} u = \Delta u , $$

where $ \Delta $ 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 $ V $ with piecewise-smooth boundary $ L $.

Let $ S $ be a bounded, complete, piecewise-smooth, two-sided (oriented) surface with piecewise-smooth boundary $ \partial S $. Then the Stokes formula will be applicable:

$$ {\int\limits \int\limits } _ { S } ( \mathbf n , \mathop{\rm rot} {\mathbf a } ) ds = \ \oint _ \partial S ( \mathbf a , \mathbf t ) dl, $$

where the vector $ \mathbf n $ normal to $ S $ and the vector $ \mathbf t $ tangent to $ \partial S $ must be determined in accordance with the orientations of the surface $ S $ and its boundary $ \partial S $. The integral $ \oint _ {\partial S } ( \mathbf a , \mathbf t ) dl $ is known as the circulation of $ \mathbf a $ along $ \partial S $. 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 $ \mathop{\rm rot} \mathbf a = 0 $. For a conservative vector field there exists the so-called scalar potential, which is a function $ v ( M) $ such that $ \mathbf a = \mathop{\rm grad} v $; here

$$ \int\limits _ { AB } ( \mathbf a , \mathbf t ) dl = v( B) - v( A), $$

where the points $ A, B \in D $, $ AB $ is a piecewise-smooth curve in $ D $, $ \mathbf t $ is the unit vector tangent to $ AB $, and $ dl $ is the line element of $ AB $.

Let the vector field $ \mathbf a ( M) $ be continuously differentiable in a bounded connected domain $ V $ with piecewise-smooth boundary $ \partial V $; the Ostrogradski formula reads as follows:

$$ {\int\limits \int\limits \int\limits } _ { V } \mathop{\rm div} \mathbf a d \sigma = \ {\int\limits \int\limits } _ \partial V ( \mathbf n , \mathbf a ) ds, $$

where $ \mathbf n $ is the exterior normal vector to $ \partial V $.

The integral $ \int \int _ {\partial V } ( \mathbf n , \mathbf a ) ds $ is said to be the flux of $ \mathbf a ( M) $ across $ \partial V $. If the flux of a vector field across an arbitrary, piecewise-smooth, non-self-intersecting, oriented surface in $ V $ which is the boundary of some bounded subdomain of $ V $ is zero, the vector field $ \mathbf a ( M) $ is said to be solenoidal in $ V $. For a continuously-differentiable vector field to be solenoidal it is necessary and sufficient that $ \mathop{\rm div} \mathbf a = 0 $ at all points of $ V $. For a solenoidal vector field $ \mathbf a ( M) $ there exists a so-called vector potential: a function $ A( M) $ such that

$$ \mathbf a = \mathop{\rm rot} A( M). $$

If the divergence and the curl of a vector field are defined at each point $ M $ of a simply-connected domain $ D $, the vector field can be represented everywhere in $ D $ as the sum of a potential field $ \mathbf a _ {1} ( M) $ and a solenoidal field $ \mathbf a _ {2} ( M) $( Helmholtz' theorem):

$$ \mathbf a ( M) = \mathbf a _ {1} ( M)+ \mathbf a _ {2} ( M). $$

Vector fields for which $ \mathop{\rm div} \mathbf a = 0 $ and $ \mathop{\rm rot} \mathbf a = 0 $ are called harmonic. The potential $ v $ of a harmonic vector field satisfies the Laplace equation. The scalar field $ v $ is also said to be harmonic. For references, see Vector calculus.

Comments

Ostrogradski's formula is commonly called Gauss' formula.

The condition $ \mathop{\rm div} a = 0 $ is necessary for a vector field to be solenoidal. It is sufficient on, for example, convex domains. The general additional condition is that the second homology of the domain vanishes. This can easily be seen from the de Rham cohomology theory. There are examples of vector fields on $ 3 $- space with one point removed which have vanishing divergence, but are not solenoidal.

The notions of gradient, divergence, Laplace operator, flux of a vector field, and the given integral formulas can easily be extended to higher-dimensional Euclidean spaces and Riemannian manifolds, and all other notions can be extended to Riemannian $ 3 $- manifolds.

In this context, the given integral formulas appear in a unified way as Stokes' formula, saying that the integral of a $ k $- form over the piecewise-regular boundary of a smooth orientable $ ( k- 1) $- submanifold is equal to the integral of its exterior differential over the submanifold itself.

References

[a1]	A. Marsden, "Calculus" , 3 , Springer (1988)
[a2]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)