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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. These three properties are equivalent in many 'real-world' applications.

Contents

Definition

Path independence

Irrotational vector fields

Irrotational fluids

Conservative forces

References

Definition

A vector field $mathbf\{v\}$ is said to be ''conservative'' if there exists a scalar field $phi$ such that
:$mathbf\{v\}=-$
ablaphi.
Here $$
ablaphi denotes the gradient of $phi$. When the above equation holds, $phi$ is called a scalar potential for $mathbf\{v\}$.

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 $Ssubseteqmathbb\{R\}^3$
is some region of three-dimensional space, and that $P$ is a path in $S$ with start point $A$ and end point $B$. If $$
mathbf{v}=-
ablaphi is a conservative vector field then
:$int\_P\; mathbf\{v\}cdot\; dmathbf\{r\}=phi(A)-phi(B).$
This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.
An equivalent formulation of this is to say that
:$oint\; mathbf\{v\}cdot\; dmathbf\{r\}=0$
for every closed loop in $S$.
The converse of the above statement is also true. That is, if the circulation of $mathbf\{v\}$ around every closed loop in $S$ 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
:$$
abla imesmathbf{v} = 0.
For this reason, such vector fields are sometimes referred to as ''curl-free'' vector fields.
It is an identity of vector calculus that for any scalar field $phi$:
:$$
abla imes
abla phi=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 $S=mathbb\{R\}^3setminus\{(0,0,z)~|~zinmathbb\{R\}\}$. Now define a vector field by
:
ight).
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 $2pi$. 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 proved directly by using Stokes' Theorem.

Irrotational fluids

The flow velocity $mathbf\{u\}$ of a fluid is a vector field, and the vorticity $mathbf\{omega\}$ of the fluid is (usually) defined by
:$mathbf\{omega\}=$
abla imesmathbf{u}.
If $mathbf\{u\}$ is irrotational then the fluid is said to be an ''irrotational fluid'', or to have ''irrotational flow''. The vorticity of an irrotational fluid 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. The gravitational force on a mass $m$ due to a mass $M$ which is a distance $r$ away can be written as
:
where $G$ is the Gravitational Constant and $hat\{mathbf\{r\}\}$ is a unit vector pointing from $M$ towards $m$. In this case $mathbf\{F\}\_G=-$
ablaphi_G, where
:
is the Gravitational potential.
In the case of 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 done in going around a closed loop is zero. In other words, the total energy of a particle moving under the influence of conservative forces is conserved.

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)