= Vector Field =

A '''vector field''' represents direction and magnitude at all points in a [[Calculus/CoordinateSystem|coordinate system]].

<<TableOfContents>>

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== 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 [[Calculus/CoordinateSystem|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'''.

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== Irrotational Fields ==

A vector field is '''irrotational''' if there is zero [[Calculus/Curl|curl]].



=== Cross-Partial Property ===

The '''cross-partial property''' is an application of '''Clairaut's theorem''' (i.e., ''f,,xy,, = f,,yx,,'') to vector-valued functions.

In two dimensions the property specifies that, given ''F = <P(x,y), Q(x,y)>'',

{{attachment:cross1.svg}}

In three dimensions is specifies that, given ''F = <P(x,y,z), Q(x,y,z), R(x,y,z)>'',

{{attachment:cross1.svg}}

{{attachment:cross2.svg}}

{{attachment:cross3.svg}}

In both cases, it should be apparent that this is equivalent to ''curl F = 0''.

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== Conservative Fields ==

A conservative field is irrotational, i.e. it has zero [[Calculus/Curl|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 ''C,,i,,'' that connect ''A'' to ''B'' and are entirely within ''D'',

{{attachment:pathind.svg}}

A conservative field ''F'' can equivalently be expressed in terms of its [[Calculus/PotentialFunction|potential function]] ''f'': ''F = ∇f''.



=== Divergence of Conservative Fields ===

[[Calculus/Divergence|Divergence]] of a conservative vector field ''F'' can be calculated as:

{{attachment:laplace1.svg}}

The ''∇^2^'' expression is the '''Laplace operator'''.

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== Source-Free Fields ==

A vector field is source-free if there is zero [[Calculus/Divergence|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 [[Calculus/FluxIntegral|flux]].

To summarize, these are tests for a source-free field:

 * {{attachment:sourcefree1.svg}}
 * {{attachment:sourcefree2.svg}} for a smooth closed curve ''C''
 * {{attachment:sourcefree3.svg}} for a vector field given as ''F=<P,Q>''
 * {{attachment:sourcefree4.svg}} 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:

{{attachment:stream1.svg}}

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== Conservative and Source-Free Fields ==

If vector field ''F'' is both conservative and source-free, then it must be that ''∇^2^f = 0''. This is '''Laplace's equation''', and a function ''f'' satisfying it is '''harmonic'''.

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