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| The ''∇^2^'' expression is the '''Laplace operator'''. |
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| If vector field ''F'' is both conservative and source-free, then the function ''f'' that satisfies ''F = ∇f'' is '''harmonic'''. | 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: |
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| A harmonic function notably satisfies Laplace's equation (''f,,xx,, + f,,y,, = 0''). | ''∇^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'''. |
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,
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.
Identifying Gradient 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)>.
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.
Derive g(x,y) + h(y) with respect to y. This produces a function like gy(x,y) + h'(y).
Set gy(x,y) + h'(y) equal to Q(x,y) and solve for h'(y).
Integrate h'(y), solving for the actual h(y).
The gradient function is g(x,y) + h(y) + C.
Laplace Operator
For a conservative F in two dimensions (x and y), the following is true:
where fxx = ∂2f/∂x2 and fyy = ∂2f/∂y2.
For a conservative F in three dimension (x, y, and z) the following is true:
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where fzz = ∂2f/∂z2.
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 circular path C, movement in is equal to movement out (flux is zero).
Conservative and Source-Free Fields
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:
∇2f = 0
which can be evaluated as fxx + fyy = 0 in two dimensions or fxx + fyy + fz = 0 in three. This is Laplace's equation.
Such a function f is harmonic.