Vector Field

A vector field represents direction and magnitude at all points in a coordinate system.

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 C_i 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)>.

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

For a conservative F in two dimensions (x and y), the following is true:

where f_xx = ∂²f/∂x² and f_yy = ∂²f/∂y².

For a conservative F in three dimension (x, y, and z) the following is true:

where f_zz = ∂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 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 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 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.

Irrotational Fields

A vector field is irrotational if there is zero curl.

CategoryRicottone