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| ---- == Source-Free Fields == A vector field is [[Calculus/VectorField#Source-Free_Fields|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 circular path C, movement in is equal to movement out (flux is zero). The last point leads to one definition of a source-free vector field: {{attachment:div3.svg}} This can be equivalently expressed as: {{attachment:div4.svg}} where ''P,,x,, = ∂P/∂x'' and ''Q,,y,, = ∂Q/∂y''. Note furthermore that this equation can only be true if ''P,,x,, + Q,,y,, = 0'', giving an even more succinct formulation. If a vector field is source-free, there is at least one '''stream function''' ''g''. If the field is given as ''F = <P(x,y), Q(x,y)>'', then ''g'' must satisfy: {{attachment:stream1.svg}} |
A vector field is [[Calculus/VectorField#Source-Free_Fields|source-free]] if there is zero divergence. |
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| A vector field is [[Calculus/VectorField#Cross-Partial_Property_of_Conservative_Fields|conservative]] if it is path independent. There is necessarily at least one function ''f'' satisfying ''F = ∇f''. In this case, divergence can be evaluated using the [[Calculus/VectorField#Laplace_Operator|Laplace operator]]. | A [[Calculus/VectorField|vector field]] is [[Calculus/VectorField#Cross-Partial_Property_of_Conservative_Fields|conservative]] if it is path independent. There is necessarily at least one function ''f'' satisfying ''F = ∇f''. In this case, divergence can be calculated using the [[Calculus/VectorField#Laplace_Operator|Laplace operator]]. For a vector field given as ''F = <P(x,y), Q(x,y)>'', applying the Laplace operator looks like: |
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| where ''f,,xx,, = ∂^2^f/∂x^2^'' and ''f,,yy,, = ∂^2^f/∂y^2^'', and | where ''f,,xx,, = ∂^2^f/∂x^2^'' and ''f,,yy,, = ∂^2^f/∂y^2^''. For a vector field given as ''F=<P,Q,R>'', applying the Laplace operator looks like: |
Divergence
Divergence measures the distortion of a vector field.
Description
The divergence of a vector field given as F = <P(x,y), Q(x,y)> is calculated as:
The divergence of F given as <P(x,y,z), Q(x,y,z), R(x,y,z)> is calculated as:
Note that divergence returns a scalar value.
Properties
The divergence of the curl of a vector field is always 0.
A vector field is source-free if there is zero divergence.
Conservative Fields
A vector field is conservative if it is path independent. There is necessarily at least one function f satisfying F = ∇f. In this case, divergence can be calculated using the Laplace operator.
For a vector field given as F = <P(x,y), Q(x,y)>, applying the Laplace operator looks like:
where fxx = ∂2f/∂x2 and fyy = ∂2f/∂y2.
For a vector field given as F=<P,Q,R>, applying the Laplace operator looks like:
where fzz = ∂2f/∂z2.