Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path, so it is possible to define a potential energy that is independent of the actual path taken.
Informal treatment.
In a two- and three-dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points—apart from the straight line formed between the two points, one could choose a curved path of greater length as shown in the figure. Therefore, in general, the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken, which can be thought of as a large-scale cancellation of all elements formula_1 that don't have a component along the straight line between the two points. To visualize this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top, they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance.
Intuitive explanation.
M. C. Escher's painting "Ascending and Descending" illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground as one moves along the staircase. It is "rotational" in that one can keep getting higher or keep getting lower while going around in circles. It is non-conservative in that one can return to one's starting point while ascending more than one descends or vice versa. On a real staircase, the height above the ground is a scalar potential field: If one returns to the same place, one goes upward exactly as much as one goes downward. Its gradient would be a conservative vector field and is irrotational. The situation depicted in the painting is impossible.
Definition.
A vector field formula_2, where formula_3 is an open subset of formula_4, is said to be conservative if and only if there exists a formula_5 scalar field formula_6 on formula_3 such that
Here, formula_9 denotes the gradient of formula_6. When the equation above holds, formula_6 is called a scalar potential for formula_12.
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 formula_12 is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that formula_14 is a rectifiable path in formula_3 with initial point formula_16 and terminal point formula_17. If formula_18 for some formula_5 scalar field formula_6 so that formula_12 is a conservative vector field, then the gradient theorem states that
This holds as a consequence of the chain rule and the fundamental theorem of calculus.
An equivalent formulation of this is that
for every rectifiable simple closed path formula_24 in formula_3. The converse of this statement is also true: If the circulation of formula_12 around every rectifiable simple closed path in formula_3 is formula_28, then formula_12 is a conservative vector field.
Irrotational vector fields.
Let formula_30, and let formula_31 be a formula_5 vector field, with formula_3 open as always. Then formula_12 is called irrotational if and only if its curl is formula_35 everywhere in formula_3, i.e., if
For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields. They are also referred to as longitudinal vector fields.
It is an identity of vector calculus that for any formula_38 scalar field formula_6 on formula_3, we have
Therefore, every formula_5 conservative vector field on formula_3 is also an irrotational vector field on formula_3.
Provided that formula_3 is simply connected, the converse of this is also true: Every irrotational vector field on formula_3 is a formula_5 conservative vector field on formula_3.
The above statement is "not" true in general if formula_3 is not simply connected. Let formula_3 be formula_51 with the formula_52-axis removed, i.e., formula_53. Now, define a vector field formula_12 on formula_3 by
Then formula_12 has zero curl everywhere in formula_3, i.e., formula_12 is irrotational. However, the circulation of formula_12 around the unit circle in the formula_61-plane is formula_62. Indeed, note that in polar coordinates, formula_63, so the integral over the unit circle is
Therefore, formula_12 does not have the path-independence property discussed above and is not conservative. 
In a simply connected open 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 simply connected open region, any vector field that has the path-independence property must also be irrotational.
More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential. The conservative vector fields correspond to the exact , that is, to the forms which are the exterior derivative formula_66 of a function (scalar field) formula_67 on formula_3. The irrotational vector fields correspond to the closed , that is, to the formula_69 such that formula_70. As any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed are exact if formula_3 is simply connected.
Vorticity.
The vorticity formula_72 of a vector field can be defined by:
The vorticity of an irrotational field is zero everywhere. Kelvin's circulation theorem states that a fluid that is irrotational in an inviscid flow will remain irrotational. This result can be derived from the vorticity transport equation, obtained by taking the curl of the Navier-Stokes Equations.
For a two-dimensional field, the vorticity acts as a measure of the "local" rotation of fluid elements. Note that the vorticity does "not" imply anything about the global behavior of a fluid. It is possible for a fluid traveling in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.
Conservative forces.
If the vector field associated to a force formula_74 is conservative, then the force is said to be a conservative force.
The most prominent examples of conservative forces are the gravitational force and the electric force associated to an electrostatic field. According to Newton's law of gravitation, the gravitational force formula_75 acting on a mass formula_76 due to a mass formula_77, which is a distance formula_78 between them, obeys the equation
where formula_80 is the gravitational constant and formula_81 is a "unit" vector pointing from formula_77 toward formula_76. The force of gravity is conservative because formula_84, where
is the gravitational potential energy. It can be shown that any vector field of the form formula_86 is conservative, provided that formula_87 is integrable.
For conservative forces, "path independence" can be interpreted to mean that the work done in going from a point formula_16 to a point formula_17 is independent of the path chosen, and that the work formula_90 done in going around a simple closed loop is formula_28:
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.