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| == Conservative Fields == | == Irrotational Fields == |
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| 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 vector field is '''irrotational''' if there is zero [[Calculus/Curl|curl]]. |
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| === Cross-Partial Property of Conservative Fields === | === Cross-Partial Property === |
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| A conservative field must satisfy the '''cross-partial property'''. | The '''cross-partial property''' is an application of '''Clairaut's theorem''' (i.e., ''f,,xy,, = f,,yx,,'') to vector-valued functions. |
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| Equivalently, test if the [[Calculus/Curl|curl]] of ''F'' is 0. | In both cases, it should be apparent that this is equivalent to ''curl F = 0''. |
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| 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. | ---- |
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| === Identifying Gradient Function === | == Conservative Fields == |
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| 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''. | 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. |
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| As an example, consider a vector field given as ''F = <P(x,y), Q(x,y)>''. | 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'', |
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| 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''. |
{{attachment:pathind.svg}} A conservative field ''F'' can equivalently be expressed in terms of its [[Calculus/PotentialFunction|potential function]] ''f'': ''F = ∇f''. |
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| === Laplace Operator === | === Divergence of Conservative Fields === |
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| For a conservative ''F'' in two dimensions (''x'' and ''y''), the following is true: | [[Calculus/Divergence|Divergence]] of a conservative vector field ''F'' can be calculated as: |
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where ''f,,xx,, = ∂^2^f/∂x^2^'' and ''f,,yy,, = ∂^2^f/∂y^2^''. For a conservative ''F'' in three dimension (''x'', ''y'', and ''z'') the following is true: {{attachment:laplace2.svg}} where ''f,,zz,, = ∂^2^f/∂z^2^''. |
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| 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 circular path ''C'', movement in is equal to movement out (flux is zero). | 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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| If vector field ''F'' is both conservative and source-free, then following from the above definition of the Laplace operator and the above property of divergence for source-free fields, it must be that: | 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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| ''∇^2^f = 0'' which can be evaluated as ''f,,xx,, + f,,yy,, = 0'' in two dimensions or ''f,,xx,, + f,,yy,, + f,,z,, = 0'' in three. This is '''Laplace's equation'''. Such a function ''f'' is '''harmonic'''. |
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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.