A
scalar potential*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
is a fundamental concept in vector analysis and
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
(the adjective
scalar is frequently omitted if there is no danger of confusion with
vector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
). The scalar potential is an example of a
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinateindependent, meaning that any two observers using the same units will agree on the...
. Given a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
F, the scalar potential
P is defined such that:
,
where
∇P is the
gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of P and the second part of the equation is minus the gradient for a function of the
Cartesian coordinatesA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
x,y,z. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of P in terms of the gradient, the direction of
F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.
In order for
F to be described in terms of a scalar potential only, the following have to be true:
 , where the integration is over a Jordan arc passing from location a to location b and P(b) is P evaluated at location b .
 , 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 gradientThe 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: \phi\left\phi\left = \int_L...
and is true for any vector field that is a gradient of a differentiable
single valuedA singlevalued function is an emphatic term for a mathematical function in the usual sense. That is, each element of the function's domain maps to a single, welldefined element of its range. This contrasts with a general binary relation, which can be viewed as being a multivalued function...
scalar field P. The second condition is a requirement of
F so that it can be expressed as the gradient of a scalar function. The third condition reexpresses the second condition in terms of the curl of
F using the
fundamental theorem of the curlIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
. A vector field
F that satisfies these conditions is said to be
irrotationalIn vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two...
(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
accelerationIn physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
due to the field, as a function of position. The gravity potential is the gravitational
potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
per unit mass. In
electrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slowmoving electric charges....
the
electric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
is the scalar potential associated with the
electric fieldIn physics, an electric field surrounds electrically charged particles and timevarying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
, i.e., with the electrostatic force per unit
chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
. The electric potential is in this case the electrostatic potential energy per unit charge. In
fluid dynamicsIn physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, irrotational lamellar fields have a scalar potential only in the special case when it is a Laplacian field. Certain aspects of the
nuclear forceThe nuclear force is the force between two or more nucleons. It is responsible for binding of protons and neutrons into atomic nuclei. The energy released causes the masses of nuclei to be less than the total mass of the protons and neutrons which form them...
can be described by a
Yukawa potential. The potential play a prominent role in the
LagrangianLagrangian mechanics is a reformulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the ItalianFrench mathematician JosephLouis Lagrange in 1788....
and
HamiltonianHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formulations of
classical mechanicsIn physics, classical mechanics is one of the two major subfields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. Further, the scalar potential is the fundamental quantity in
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Not every vector field has a scalar potential. Those that do are called
conservative, corresponding to the notion of
conservative forceA conservative force is a force with the property that the 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 net work done by a conservative force is zero.It is possible to define a numerical value of...
in physics. Examples of nonconservative forces include frictional forces, magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the
Helmholtz decompositionIn 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 the sum of an irrotational vector field and a...
theorem however, all vector fields can be describable in terms of a scalar potential and corresponding
vector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
. In electrodynamics the electromagnetic scalar and vector potentials are known together as the
electromagnetic fourpotentialThe electromagnetic fourpotential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single spacetime fourvector. In a given reference frame, the first component is the scalar potential and...
.
Integrability conditions
If
F is a conservative vector field (also called
irrotational,
curlfree, or
potential), and its components have
continuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
partial derivativeIn 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...
s, the
potential of
F with respect to a reference point
is defined in terms of the
line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
:
where
C is a parametrized path from
to

The fact that the line integral depends on the path
C only through its terminal points
and
is, in essence, the
path independence property of a conservative vector field. The
fundamental theorem of calculusThe first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
for line integrals implies that if
V is defined in this way, then
so that
V is a scalar potential of the conservative vector field
F. 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
V is defined in terms of the line integral, the ambiguity of
V reflects the freedom in the choice of the reference point
Altitude as gravitational potential energy
An example is the (nearly) uniform
gravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
near the Earth's surface. It has a potential energy
where
U is the gravitational potential energy and
h is the height above the surface. This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the twodimensional negative gradient of the altitude is a twodimensional 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 threedimensional negative gradient of
U always points straight downwards in the direction of gravity;
F. 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
θ is the angle of inclination, and the component of
F_{S} perpendicular to gravity is
This force
F_{P}, parallel to the ground, is greatest when
θ is 45 degrees.
Let Δ
h be the uniform interval of altitude between contours on the contour map, and let Δ
x be the distance between two contours. Then
so that
However, on a contour map, the gradient is inversely proportional to Δ
x, which is not similar to force
F_{P}: altitude on a contour map is not exactly a twodimensional 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 mechanicsFluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, 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 forceIn physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...
is the negative gradient of
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
:
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 ground. The surface of the water can be characterized as a plane with zero pressure.
If the liquid has a vertical
vortexA vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
(whose axis of rotation is perpendicular to the ground), then the vortex causes a depression in the pressure field. The surfaces of constant pressure are parallel to the ground far away from the vortex, but near and inside the vortex the surfaces of constant pressure are pulled downwards, closer to the ground. This also happens to the surface of zero pressure. Therefore, inside the vortex, the top surface of the liquid is pulled downwards into a depression, or even into a tube (a solenoid).
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:
A moving airplane wing makes the air pressure above it decrease relative to the air pressure below it. This creates enough buoyant force to counteract gravity.
Calculating the scalar potential
Given a vector field
E, its scalar potential
Φ can be calculated to be
where
τ is volume. Then, if
E is
irrotationalIn vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two...
(Conservative),
This formula is known to be correct if
E is
continuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
and vanishes asymptotically to zero towards infinity, decaying faster than 1/
r and if the
divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of
E likewise vanishes towards infinity, decaying faster than 1/
r^{2}.
See also
 Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
 Fundamental theorem of vector analysis
 Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
 Yukawa potential