In
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, a conservative vector field is a
vector field that is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of some
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
. A conservative vector field has the property that its
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
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 under the line integral being conservative. A conservative vector field is also
irrotational
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 any path between two points does not c ...
; in three dimensions, this means that it has vanishing
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
. An irrotational vector field is necessarily conservative provided that the domain is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
.
Conservative vector fields appear naturally in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
: They are vector fields representing
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s of
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s in which
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
is
conserved. For a conservative system, the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
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
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
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
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.
Intuitive explanation
M. C. Escher's lithograph print ''
Ascending and Descending
''Ascending and Descending'' is a lithograph print by the Dutch artist M. C. Escher first printed in March 1960. The original print measures . The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identicall ...
'' 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 print is impossible.
Definition
A
vector field , where
is an open subset of
, is said to be conservative if and only if there exists a
(
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
)
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
[For to be path-independent, is not necessarily continuously differentiable, the condition of being differentiable is enough, since the ]Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
, that proves the path independence of , does not require to be continuously differentiable.
There must be a reason for the definition of conservative vector fields to require to be continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. on
such that
Here,
denotes the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of
. Since
is continuously differentiable,
is continuous. When the equation above holds,
is called a
scalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
for
.
The
fundamental theorem of vector calculus
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth function, smooth, rapidly decaying vector field in three dimensions can b ...
states that any vector field can be expressed as the sum of a conservative vector field and a
solenoidal field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a Helmholtz decomposition#Longitudinal and transverse fields, transverse vector field) is a vector field v with diverge ...
.
Path independence and conservative vector field
Path independence
A line integral of a vector field
is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:
for any pair of integral paths
and
between a given pair of path endpoints in
.
The path independence is also equivalently expressed as
for any piecewise smooth closed path
in
where the two endpoints are coincident. Two expressions are equivalent since any closed path
can be made by two path;
from an endpoint
to another endpoint
, and
from
to
, so
where
is the reverse of
and the last equality holds due to the path independence
Conservative vector field
A key property of a 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.'' Suppose that
for some
(
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
) scalar field
over
as an open subset of
(so
is a conservative vector field that is continuous) and
is a differentiable path (i.e., it can be parameterized by a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
) in
with an initial point
and a terminal point
. Then the
gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
(also called ''fundamental theorem of calculus for line integrals'') states that
This holds as a consequence of the
definition of a line integral, the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, and the
second fundamental theorem of calculus.
in the line integral is an
exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&nbs ...
for an orthogonal coordinate system (e.g.,
Cartesian Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to:
Mathematics
*Cartesian closed category, a closed category in category theory
*Cartesian coordinate system, modern ...
,
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...
, or
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
). 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.
So far it has been proven that a conservative vector field
is line integral path-independent. Conversely, ''if a continuous vector field
is (line integral) path-independent, then it is a conservative vector field'', so the following
biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as t ...
statement holds:
The proof of this converse statement is the following.
is a continuous vector field which line integral is path-independent. Then, let's make a function
defined as
over an arbitrary path between a chosen starting point
and an arbitrary point
. Since it is path-independent, it depends on only
and
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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
is used. The second segment of this path is parallel to the
axis so there is no change along the
axis. The line integral along this path is
By the path independence, its partial derivative with respect to
(for
to have partial derivatives,
needs to be continuous.) is
since
and
are independent to each other. Let's express
as
where
and
are unit vectors along the
and
axes respectively, then, since
,
where the last equality is from the
second fundamental theorem of calculus.
A similar approach for the line integral path shown in the right of the right figure results in
so
is proved for the 2-dimensional
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensional
spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
) so the converse statement is proved. Another proof is found
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here
Television
* Here TV (form ...
as the converse of the gradient theorem.
Irrotational vector fields
Let
(3-dimensional space), and let
be a
(
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
) vector field, with an open subset
of
. Then
is called irrotational if and only if its
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
is
everywhere in
, 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
(
continuously differentiable up to the 2nd derivative) scalar field
on
, we have
Therefore, ''every
conservative vector field in
is also an irrotational vector field in
''. This result can be easily proved by expressing
in a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
with
Schwarz's theorem
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n'' ...
(also called Clairaut's theorem on equality of mixed partials).
Provided that
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: ''Every irrotational vector field in a simply connected open space
is a
conservative vector field in
''.
The above statement is ''not'' true in general if
is not simply connected. Let
be
with removing all coordinates on the
-axis (so not a simply connected space), i.e.,
. Now, define a vector field
on
by
Then
has zero curl everywhere in
(
at everywhere in
), i.e.,
is irrotational. However, the
circulation of
around the unit circle in the
-plane is
; in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
,
, so the integral over the unit circle is
Therefore,
does not have the path-independence property discussed above so is not conservative even if
since
where
is defined is not a simply connected open space.
Say again, in a simply connected open region, an irrotational vector field
has the path-independence property (so
as conservative). This can be proved directly by using
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
,
for any smooth oriented surface
which boundary is a simple closed path
. 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 vise versa.''
Abstraction
More abstractly, in the presence of a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
, vector fields correspond to
differential . The conservative vector fields correspond to the
exact
Exact may refer to:
* Exaction, a concept in real property law
* ''Ex'Act'', 2016 studio album by Exo
* Schooner Exact, the ship which carried the founders of Seattle
Companies
* Exact (company), a Dutch software company
* Exact Change, an Ameri ...
, that is, to the forms which are the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
of a function (scalar field)
on
. The irrotational vector fields correspond to the
closed , that is, to the
such that
. As any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed are exact if
is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
.
Vorticity
The
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
of a vector field can be defined by:
The vorticity of an irrotational field is zero everywhere.
[, pp. 194–196.] Kelvin's circulation theorem
In fluid mechanics, Kelvin's circulation theorem (named after William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and ...
states that a fluid that is irrotational in an
inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as ...
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 to travel in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.
Conservative forces
![Conservative fields](https://upload.wikimedia.org/wikipedia/commons/9/9d/Conservative_fields.png)
If the vector field associated to a force
is conservative, then the force is said to be a
conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum o ...
.
The most prominent examples of conservative forces are a gravitational force and an electric force associated to an electrostatic field. According to
Newton's law of gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
, a
gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
acting on a mass
due to a mass
located at a distance
from
, obeys the equation
where
is the
gravitational constant
The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
and
is a ''unit'' vector pointing from
toward
. The force of gravity is conservative because
, where
is the
gravitational potential energy
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conv ...
. In other words, the gravitation field
associated with the gravitational force
is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of the gravitation potential
associated with the gravitational potential energy
. It can be shown that any vector field of the form
is conservative, provided that
is integrable.
For
conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum o ...
s, ''path independence'' can be interpreted to mean that the
work done
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
in going from a point
to a point
is independent of the moving path chosen (dependent on only the points
and
), and that the work
done in going around a simple closed loop
is
:
The total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
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 In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that
\mathbf\times (\nabla\times\mathbf)=0.
Thus \m ...
*
Conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum o ...
*
Conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink ...
*
Complex lamellar vector field In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector ...
*
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
*
Laplacian vector field
In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:
:\begin
\nabla \times \mathbf &= \mathbf ...
*
Longitudinal and transverse vector fields
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
*
Solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
References
Further reading
* {{cite book , first= D. J. , last= Acheson , title= Elementary Fluid Dynamics , publisher= Oxford University Press , date= 1990 , isbn= 0198596790
Vector calculus
Force