Potential field

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gradient field

The vector field generated by the gradients of a scalar function in several variables which belong to some domain in an -dimensional space. The function is called the scalar potential (potential function) of this field. A potential field is completely integrable over : The Pfaffian equation has the level lines or the level surfaces of the potential as -dimensional integral manifolds. Any regular covariant field that is completely integrable over is obtained by multiplying the potential field by a scalar:

The scalar is called an integrating factor of the Pfaffian equation . The following equalities serve as a test for the field to be the gradient of a potential ():

They indicate that the field is rotation free (see Curl).

The concept of a potential field is widely used in mechanics and physics. The majority of force fields and electric fields can be considered as potential fields. For instance, if is the pressure at a point of an ideal fluid filling a region , then the vector is equal to the equilibrium pressure force applied to the volume element . If is the temperature of a heated body at a point , then the vector , where is the thermal conductivity coefficient, is equal to the density of the heat flow in the direction of less heated parts of the body (in the direction orthogonal to the isothermal surfaces ).


In the above, complete integrability of a vector field means that the Pfaffian equation defines an involutive distribution, i.e. an integrable one. A differential such that for some potential is called a total differential and the corresponding function is sometimes called a complete integral. Especially for , is called an exact differential equation.


[a1] H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. Sect. 10.1.4
[a2] E. Zauderer, "Partial differential equations" , Wiley (Interscience) (1989) pp. 92
[a3] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. Sects. 12.3, 14.7
How to Cite This Entry:
Potential field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article