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
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 <math>d{R}</math> 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.
thumbnail|Depiction of two possible paths to integrate. In green is the simplest possible path; blue shows a more convoluted curve
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 <math>\mathbf{v}: U \to \R^n</math>, where <math>U</math> is an open subset of <math>\R^n</math>, is said to be conservative if there exists a <math>C^1</math> (continuously differentiable) scalar field <math>\varphi</math> on <math>U</math> such that
<math display="block">\mathbf{v} = \nabla \varphi.</math>
Here, <math>\nabla \varphi</math> denotes the gradient of <math>\varphi</math>. Since <math>\varphi</math> is continuously differentiable, <math>\mathbf{v}</math> is continuous. When the equation above holds, <math>\varphi</math> is called a scalar potential for <math>\mathbf{v}</math>.
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
A line integral of a vector field <math>\mathbf{v}</math> is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:
<math display="block">\int_{P_1} \mathbf{v} \cdot d \mathbf{r} = \int_{P_2} \mathbf{v} \cdot d \mathbf{r}</math>
for any pair of integral paths <math>P_1</math> and <math>P_2</math> between a given pair of path endpoints in <math>U</math>.
The path independence is also equivalently expressed as
<math display="block">\int_{P_c} \mathbf{v} \cdot d \mathbf{r} = 0</math>
for any piecewise smooth closed path <math>P_c</math> in <math>U</math> where the two endpoints are coincident. Two expressions are equivalent since any closed path <math>P_c</math> can be made by two path; <math>P_1</math> from an endpoint <math>A</math> to another endpoint <math>B</math>, and <math>P_2</math> from <math>B</math> to <math>A</math>, so
<math display="block">\int_{P_c} \mathbf{v} \cdot d \mathbf{r} = \int_{P_1} \mathbf{v} \cdot d \mathbf{r} + \int_{P_2} \mathbf{v} \cdot d \mathbf{r} = \int_{P_1} \mathbf{v} \cdot d \mathbf{r} - \int_{-P_2} \mathbf{v} \cdot d \mathbf{r} = 0</math>
where <math>-P_2</math> is the reverse of <math>P_2</math> and the last equality holds due to the path independence <math display="inline">\displaystyle \int_{P_1} \mathbf{v} \cdot d \mathbf{r} = \int_{-P_2} \mathbf{v} \cdot d \mathbf{r}.</math>
Conservative vector field
A key property of a conservative vector field <math>\mathbf{v}</math> 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. Suppose that <math>\mathbf{v} = \nabla \varphi</math> for some <math>C^1</math> (continuously differentiable) scalar field <math>\varphi</math>
So far it has been proven that a conservative vector field <math>\mathbf{v}</math> is line integral path-independent. Conversely, if a continuous vector field <math>\mathbf{v}</math> is (line integral) path-independent, then it is a conservative vector field, so the following biconditional statement holds: 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
[[File:Conservative fields.png|thumb|upright=1.5|Examples of potential and gradient fields in physics:
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If the vector field associated to a force <math>\mathbf{F}</math> 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 <math>\mathbf{F}_{G}</math> acting on a mass <math>m</math> due to a mass <math>M</math> located at a distance <math>r</math> from <math>m</math>, obeys the equation
<math display="block">\mathbf{F}_{G} = - \frac{G m M}{r^2} \hat{\mathbf{r,</math>
where <math>G</math> is the gravitational constant and <math>\hat{\mathbf{r</math> is a unit vector pointing from <math>M</math> toward <math>m</math>. The force of gravity is conservative because <math>\mathbf{F}_{G} = - \nabla \Phi_{G}</math>, where
<math display="block">\Phi_{G} ~ \stackrel{\text{def{=} - \frac{G m M}{r}</math>
is the gravitational potential energy. In other words, the gravitation field <math>\frac{\mathbf{F}_{G{m}</math> associated with the gravitational force <math>\mathbf{F}_{G}</math> is the gradient of the gravitation potential <math>\frac{\Phi_{G{m}</math> associated with the gravitational potential energy <math>\Phi_{G}</math>. It can be shown that any vector field of the form <math>\mathbf{F}=F(r) \hat{\mathbf{r</math> is conservative, provided that <math>F(r)</math> is integrable.
For conservative forces, path independence can be interpreted to mean that the work done in going from a point <math>A</math> to a point <math>B</math> is independent of the moving path chosen (dependent on only the points <math>A</math> and <math>B</math>), and that the work <math>W</math> done in going around a simple closed loop <math>C</math> is <math>0</math>:
<math display="block">W = \oint_{C} \mathbf{F} \cdot d{\mathbf{r = 0.</math>
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.
See also
- Beltrami vector field
- Conservative force
- Conservative system
- Complex lamellar vector field
- Helmholtz decomposition
- Laplacian vector field
- Longitudinal and transverse vector fields
- Solenoidal vector field