Vector Field
A vector field represents direction and magnitude at all points in a coordinate system.
Contents
Description
A vector field F is defined for some domain D and maps points to a vector. Generally the vectors have as many dimensions as the coordinate system. That is, a point (x,y) maps to a vector <P,Q>; a point (x,y,z) maps to a vector <P,Q,R>.
A vector field's domain can be characterized as simply connected, connected, or not connected.
Irrotational Fields
A vector field is irrotational if there is zero curl.
Cross-Partial Property
The cross-partial property is an application of Clairaut's theorem (i.e., fxy = fyx) to vector-valued functions.
In two dimensions the property specifies that, given F = <P(x,y), Q(x,y)>,
In three dimensions is specifies that, given F = <P(x,y,z), Q(x,y,z), R(x,y,z)>,
In both cases, it should be apparent that this is equivalent to curl F = 0.
Conservative Fields
A conservative field is irrotational, i.e. it has zero curl. The distinction is that a conservative field must also be simply connected.
As a consequence, a conservative field is path independent. That is, for all Ci that connect A to B and are entirely within D,
A conservative field F can equivalently be expressed in terms of its potential function f: F = ∇f.
Divergence of Conservative Fields
Divergence of a conservative vector field F can be calculated as:
The ∇2 expression is the Laplace operator.
Source-Free Fields
A vector field is source-free if there is zero divergence. This means that there are no sources (points where the field originates) or sinks (points where the field terminates). A consequence is that, for any closed curve or surface, there is zero flux.
To summarize, these are tests for a source-free field:
for a smooth closed curve C 
for a vector field given as F=<P,Q> 
for a vector field given as F=<P,Q>
If a vector field is source-free, there is at least one stream function g. If the field is given as F=<P,Q>, then g must satisfy:
Conservative and Source-Free Fields
If vector field F is both conservative and source-free, then it must be that ∇2f = 0. This is Laplace's equation, and a function f satisfying it is harmonic.