Conservative Vector Field - Irrotational Vector Fields

Irrotational Vector Fields

A vector field is said to be irrotational if its curl is zero. That is, if

For this reason, such vector fields are sometimes referred to as curl-free or curl-less vector fields.

It is an identity of vector calculus that for any scalar field :

Therefore every conservative vector field is also an irrotational vector field.

Provided that is a simply-connected region, the converse of this is true: every irrotational vector field is also a conservative vector field.

The above statement is not true if is not simply-connected. Let be the usual 3-dimensional space, except with the -axis removed; that is . Now define a vector field by

Then exists and has zero curl at every point in ; that is is irrotational. However the circulation of around the unit circle in the -plane is equal to . Indeed we note that in polar coordinates, so the integral over the unit circle is equal . Therefore does not have the path independence property discussed above, and is not conservative. (However, in any connected subregion of S, it is still true that it is conservative. In fact, the field above is the gradient of . As we know from complex analysis, this is a multi-valued function which requires a branch cut from the origin to infinity to be defined in a continuous way; hence, in a region that does not go around the z-axis, its gradient is conservative.)

In a simply-connected region an irrotational vector field has the path independence property. This can be seen by noting that in such a region an irrotational vector field is conservative, and conservative vector fields have the path independence property. The result can also be proved directly by using Stokes' theorem. In a connected region any vector field which has the path independence property must also be irrotational.

More abstractly, a conservative vector field is an exact 1-form. That is, it is a 1-form equal to the exterior derivative of some 0-form (scalar field) . An irrotational vector field is a closed 1-form. Since d2 = 0, any exact form is closed, so any conservative vector field is irrotational. The domain is simply connected if and only if its first homology group is 0, which is equivalent to its first cohomology group being 0. The first de Rham cohomology group is 0 if and only if all closed 1-forms are exact.