Conservative vector field

In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields. Every conservative vector field has zero curl (and is thus irrotational), and every conservative vector field has the path independence property. In fact, these three properties are equivalent in many 'real-world' applications.

Definition

A vector field $\mathbf {v}$ is said to be conservative if there exists a scalar field $\varphi$ such that

\mathbf {v} =\nabla \varphi .

Here $\nabla \varphi$ denotes the gradient of $\varphi$ . When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf {v}$ . The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.

Path independence

A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that $S\subseteq \mathbb {R} ^{3}$ is a region of three-dimensional space, and that $P$ is a path in $S$ with start point $A$ and end point $B$ . If $\mathbf {v} =\nabla \varphi$ is a conservative vector field then

\int _{P}\mathbf {v} \cdot d\mathbf {r} =\varphi (A)-\varphi (B).

This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.

An equivalent formulation of this is to say that

\oint \mathbf {v} \cdot d\mathbf {r} =0

for every closed loop in $S$ .

The converse of the above statement is also true provided that $S$ is a connected region. That is, if the circulation of $\mathbf {v}$ around every closed loop in a connected region $S$ is zero, then $\mathbf {v}$ is a conservative vector field.

Irrotational vector fields

A vector field $\mathbf {v}$ is said to be irrotational if its curl is zero. That is, if

\nabla \times \mathbf {v} =0.

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

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

\nabla \times \nabla \varphi =0.

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

Provided that $S$ 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 $S$ is not simply-connected. Let $S$ be the usual 3-dimensional space, except with the $z$ -axis removed; that is $S=\mathbb {R} ^{3}\setminus \{(0,0,z)~|~z\in \mathbb {R} \}$ . Now define a vector field by

\mathbf {v} =\left({\frac {-y}{x^{2}+y^{2}}},{\frac {x}{x^{2}+y^{2}}},0\right).

Then $\mathbf {v}$ exists and has zero curl at every point in $S$ ; that is $\mathbf {v}$ is irrotational. However the circulation of $\mathbf {v}$ around the unit circle in the $x,y$ -plane is equal to $2\pi$ . Therefore $v$ does not have the path independence property discussed above, and is not 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) $\phi$ . 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 $H_{\mathrm {dR} }^{1}$ is 0 if and only if all closed 1-forms are exact.

Irrotational flows

The flow velocity $\mathbf {u}$ of a fluid is a vector field, and the vorticity ${\boldsymbol {\omega }}$ of the flow is (usually) defined by

{\boldsymbol {\omega }}=\nabla \times \mathbf {u} .

If $\mathbf {u}$ is irrotational then the flow is said to be an irrotational flow. The vorticity of an irrotational flow is zero.

For a two-dimensional flow the vorticity acts as a measure of the local rotation of fluid elements. Note that the vorticity does not imply anything about the global behaviour of a fluid. It is possible for a fluid traveling in a straight line to have vorticity, and it is possible for a fluid which moves in a circle to be irrotational. For more information see: Vortex.

Conservative forces

If the vector field associated to a force $\mathbf {F}$ is conservative then the force is said to be a conservative force.

The most prominent example of a conservative force is the force of gravity. According to Newton's law of gravitation, the gravitational force, $\mathbf {F} _{G}$ , acting on a mass $m$ , due to a mass $M$ which is a distance $r$ away, obeys the equation

\mathbf {F} _{G}=-{\frac {GmM{\hat {\mathbf {r} }}}{r^{2}}},

where $G$ is the Gravitational Constant and ${\hat {\mathbf {r} }}$ is a unit vector pointing from $M$ towards $m$ . The force of gravity is conservative because $\mathbf {F} _{G}=-\nabla \Phi _{G}$ , where

\Phi _{G}=-{\frac {GmM}{r}}

is the Gravitational potential.

For a conservative forces, path independence can be interpreted to mean that the work done in going from a point $A$ to a point $B$ is independent of the path chosen, and that the work W done in going around a closed loop is zero:

W=\oint \mathbf {F} \cdot d\mathbf {r} =0.

The total energy of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to an equal quantity of kinetic energy or vice versa.

References

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press (2005)
D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press (2005)