In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, 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 traveling from one position to the other. It is a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
in
three-space: a directionless value (
scalar) that depends only on its location. A familiar example is
potential energy due to gravity.
A ''scalar
potential'' is a fundamental concept in
vector analysis
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 subjec ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
(the adjective ''scalar'' is frequently omitted if there is no danger of confusion with ''
vector potential''). The scalar potential is an example of a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
. Given a
vector field , the scalar potential is defined such that:
:
where 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 gr ...
of and the second part of the equation is minus the gradient for a function of the
Cartesian coordinates
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 ...
. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential.
[Se]
for an example where the potential is defined without a negative. Other references such as avoid using the term ''potential'' when solving for a function from its gradient. Because of this definition of in terms of the gradient, the direction of at any point is the direction of the steepest decrease of at that point, its magnitude is the rate of that decrease per unit length.
In order for to be described in terms of a scalar potential only, any of the following equivalent statements have to be true:
#
where the integration is over a
Jordan arc passing from location to location and is evaluated at location .
#
where the integral is over any simple closed path, otherwise known as a
Jordan curve.
#
The first of these conditions represents the
fundamental theorem of the gradient and is true for any vector field that is a gradient of a
differentiable single valued scalar field . The second condition is a requirement of so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the
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 ...
of using the
fundamental theorem of the curl. A vector field that satisfies these conditions is said to be
irrotational (conservative).
Scalar potentials play a prominent role in many areas of physics and engineering. The
gravity potential is the scalar potential associated with the gravity per unit mass, i.e., the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
due to the field, as a function of position. The gravity potential is the
gravitational potential energy per unit mass. In
electrostatics
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
is the scalar potential associated with the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
, i.e., with the
electrostatic force per unit
charge. The electric potential is in this case the electrostatic potential energy per unit charge. In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, irrotational
lamellar 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 ...
s have a scalar potential only in the special case when it is a
Laplacian field. Certain aspects of the
nuclear force can be described by a
Yukawa potential. The potential play a prominent role in the
Lagrangian and
Hamiltonian formulations of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. Further, the scalar potential is the fundamental quantity in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.
Not every vector field has a scalar potential. Those that do are called ''
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization in ...
'', corresponding to the notion of
conservative force in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a
solenoidal 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 \mathb ...
velocity field. By the
Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding
vector potential. In electrodynamics, the electromagnetic scalar and vector potentials are known together as the
electromagnetic four-potential.
Integrability conditions
If is a
conservative vector field (also called ''irrotational'', ''
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 ...
-free'', or ''potential''), and its components have
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s, the potential of with respect to a reference point is defined in terms of the
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, ...
:
:
where is a
parametrized path from to ,
:
The fact that the line integral depends on the path only through its terminal points and is, in essence, the path independence property of a conservative vector field. The
fundamental theorem of line integrals
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 ...
implies that if is defined in this way, then , so that is a scalar potential of the conservative vector field . Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If is defined in terms of the line integral, the ambiguity of reflects the freedom in the choice of the reference point .
Altitude as gravitational potential energy
An example is the (nearly) uniform
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational pheno ...
near the Earth's surface. It has a potential energy
:
where is the gravitational potential energy and is the height above the surface. This means that gravitational potential energy on a
contour map
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph ...
is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of always points straight downwards in the direction of gravity; . However, a ball rolling down a hill cannot move directly downwards due to the
normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:
:
where is the angle of inclination, and the component of perpendicular to gravity is
:
This force , parallel to the ground, is greatest when is 45 degrees.
Let be the uniform interval of altitude between contours on the contour map, and let be the distance between two contours. Then
so that
However, on a contour map, the gradient is inversely proportional to , which is not similar to force : altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.
Pressure as buoyant potential
In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This
buoyant force
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
is the negative gradient of
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
:
:
Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the surface, which can be characterized as the plane of zero pressure.
If the liquid has a vertical
vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...
(whose axis of rotation is perpendicular to the surface), then the vortex causes a depression in the pressure field. The surface of the liquid inside the vortex is pulled downwards as are any surfaces of equal pressure, which still remain parallel to the liquids surface. The effect is strongest inside the vortex and decreases rapidly with the distance from the vortex axis.
The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative pressure gradient along the surface of the object:
:
Scalar potential in Euclidean space
In 3-dimensional Euclidean space , the scalar potential of an
irrotational vector field is given by
where is an infinitesimal volume element with respect to . Then
This holds provided is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
and vanishes asymptotically to zero towards infinity, decaying faster than and if the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of likewise vanishes towards infinity, decaying faster than .
Written another way, let
be the
Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...
. This is the
fundamental solution of the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
, meaning that the Laplacian of is equal to the negative of the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
:
:
Then the scalar potential is the divergence of the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of with :
:
Indeed, convolution of an irrotational vector field with a rotationally
invariant potential is also irrotational. For an irrotational vector field , it can be shown that
Hence
as required.
More generally, the formula
holds in -dimensional Euclidean space () with the Newtonian potential given then by
where is the volume of the unit -ball. The proof is identical. Alternatively, integration by parts (or, more rigorously, the
properties of convolution) gives
See also
*
Gradient theorem
*
Fundamental theorem of vector analysis
*
Equipotential (isopotential) lines and surfaces
Notes
References
{{DEFAULTSORT:Scalar Potential
Potentials
Vector calculus
Scalar physical quantities
fr:Champ de vecteurs#Champ de gradient