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| A conservative field ''F'' can equivalently be expressed in terms of its [[Calculus/PotentialFunction|potential function]] ''f'': ''F = ∇f''. |
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| === Identifying the Field's Function === if a vector field is conservative, then there exists at least one function ''f'' satisfying ''F = ∇f''. There is a general process for identifying this function ''f''. As an example, consider a vector field given as ''F = <P(x,y), Q(x,y)>''. 1. Integrate ''P'' with respect to ''x''. This produces a function like ''g(x,y) + h(y)'' where ''h(y)'' is unknown, encompasses the constant ''C'', and accounts for variation in all variables that were held constant for the partial integration. 2. Derive ''g(x,y) + h(y)'' with respect to ''y''. This produces a function like ''g,,y,,(x,y) + h'(y)''. 3. Set ''g,,y,,(x,y) + h'(y)'' equal to ''Q(x,y)'' and solve for ''h'(y)''. 4. Integrate ''h'(y)'', solving for the actual ''h(y)''. 5. The gradient function is ''g(x,y) + h(y) + C''. === Laplace Operator === |
=== Divergence of Conservative Fields === |
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.
Conservative Fields
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.
Cross-Partial Property of Conservative Fields
A conservative field must satisfy the cross-partial property.
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)>,
Equivalently, test if the curl of F is 0.
If the domain D is simply connected, then satisfying this property is enough to confirm that a vector field is conservative. Otherwise there are more edge cases to consider.
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.
Irrotational Fields
A vector field is irrotational if there is zero curl.