Conservative vector field on arvau

Wikipedia / Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of 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 under the line integral 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 a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual path taken.

Informal treatment

that do not 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.

Intuitive explanation

M. C. Escher's lithograph print Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can return to the starting point while ascending more than one descends or vice versa, resulting in nonzero work done by gravity. On a real staircase, the height above the ground is a scalar potential field: one has to go upward exactly as much as one goes downward in order to return to the same place, in which case the work by gravity totals to zero. This suggests path-independence of work done on the staircase; equivalently, the force field experienced is conservative (see the later section: Path independence and conservative vector field). The situation depicted in the print is impossible.

Definition

A vector field (continuously differentiable) scalar field such that

denotes the gradient is called a scalar potential.

The fundamental theorem of vector calculus states that, under some regularity conditions, any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.

Path independence and conservative vector field

Path independence

is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:

The path independence is also equivalently expressed as

for any piecewise

Conservative vector field

is that its integral along a path depends on only the endpoints of that path, not the particular route taken. In other words, if it is a conservative vector field, then its line integral is path-independent. (continuously differentiable is a differentiable path (i.e., it can be parameterized by a differentiable function. Then the gradient theorem (also called fundamental theorem of calculus for line integrals

This holds as a consequence of the definition of a line integral, the chain rule, and the second fundamental theorem of calculus in the line integral is an exact differential for an orthogonal coordinate system (e.g., Cartesian, cylindrical, or spherical coordinates). Since the gradient theorem is applicable for a differentiable path, the path independence of a conservative vector field over piecewise-differential curves is also proved by the proof per differentiable curve component.

is line integral path-independent. Conversely, is (line integral) path-independent, then it is a conservative vector field, so the following biconditional statement holds: The proof of this converse statement is the following. regardless of which path between these points is chosen.

Let's choose the path shown in the left of the right figure where a 2-dimensional Cartesian coordinate system By the path independence, its partial derivative where the last equality is from the second fundamental theorem of calculus.

is proved for the 2-dimensional Cartesian coordinate system. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensional spherical coordinate system) so the converse statement is proved. Another proof is found here as the converse of the gradient theorem.

Irrotational vector fields

(continuously differentiable is called irrotational if its curl

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 (continuously differentiable up to the 2nd derivative

Therefore, in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials).

is a simply connected open space (roughly speaking, a single piece open space without a hole within it), the converse of this is also true: .

The above statement is not-axis (so not a simply connected space

is irrotational. However, the circulation around the unit circle; in polar coordinates

is defined is not a simply connected open space.

as conservative). This can be proved directly by using Stokes' theorem. So, it is concluded that In a simply connected open region, any vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vice versa.

Abstraction

More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential 1-forms. The conservative vector fields correspond to the exact , that is, to the forms which are the exterior derivative. The irrotational vector fields correspond to the closed , that is, to the . As any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed is simply connected.

Vorticity

The vorticity

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. The vorticity does not imply anything about the global behavior of a fluid. It is possible for a fluid that travels in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.

Conservative forces

is conservative, then the force is said to be a conservative force.

The most prominent examples of conservative forces are gravitational force (associated with a gravitational field) and electric force (associated with an electrostatic field). According to Newton's law of gravitation, a gravitational force, obeys the equation

is the gravitational constant is a unit, where

is the gravitational potential energy is the gradient is integrable.

For conservative forces, path independence can be interpreted to mean that the work done:

The total mechanical 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 the equal quantity of kinetic energy, or vice versa.