Potential field

gradient field

The vector field generated by the gradients of a scalar function $ f $ in several variables $ t = ( t ^ {1} \dots t ^ {n} ) $ which belong to some domain $ T $ in an $ n $- dimensional space. The function $ f $ is called the scalar potential (potential function) of this field. A potential field is completely integrable over $ T $: The Pfaffian equation $ ( \mathop{\rm grad} f ( t) , d t ) = 0 $ has the level lines $ ( n = 2 ) $ or the level surfaces $ ( n \geq 3 ) $ of the potential $ f $ as $ ( n - 1 ) $- dimensional integral manifolds. Any regular covariant field $ v = ( v _ {1} \dots v _ {n} ) $ that is completely integrable over $ T $ is obtained by multiplying the potential field by a scalar:

$$ v _ \alpha ( t) = c ( t) \frac{\partial f }{\partial t ^ \alpha } ,\ \ 1 \leq \alpha \leq n . $$

The scalar $ 1 / c ( t) $ is called an integrating factor of the Pfaffian equation $ ( v ( t) , d t ) = 0 $. The following equalities serve as a test for the field $ v _ \alpha ( t) $ to be the gradient of a potential ( $ c ( t) = 1 $):

$$ \frac{\partial v _ \alpha }{\partial t ^ \beta } = \ \frac{\partial v _ \beta }{\partial t ^ \alpha } ,\ \ 1 \leq \alpha , \beta \leq n . $$

They indicate that the field $ v ( t) $ 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 $ f ( t) $ is the pressure at a point $ t $ of an ideal fluid filling a region $ T $, then the vector $ F = - \mathop{\rm grad} f \cdot d \omega $ is equal to the equilibrium pressure force applied to the volume element $ d \omega $. If $ f ( t) $ is the temperature of a heated body $ T $ at a point $ t $, then the vector $ F = - k \cdot \mathop{\rm grad} f $, where $ k $ 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 $ f = \textrm{ const } $).

Comments

In the above, complete integrability of a vector field $ ( v _ {1} \dots v _ {n} ) $ means that the Pfaffian equation $ v _ {1} dt ^ {1} + \dots + v _ {n} dt ^ {n} = 0 $ defines an involutive distribution, i.e. an integrable one. A differential $ v _ {1} dt ^ {1} + \dots + v _ {n} dt ^ {n} $ such that $ v _ {i} = {\partial f } / {\partial t ^ {i} } $ for some potential $ f $ is called a total differential and the corresponding function $ f $ is sometimes called a complete integral. Especially for $ n = 2 $, $ v _ {1} dt ^ {1} + v _ {2} dt ^ {2} = 0 $ is called an exact differential equation.

References

[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