gradient theorem

If  u=u(x,y,z)  is continuously differentiable function in a simply connected domain ( D of 3 and  P=(x0,y0,z0)  and  Q=(x1,y1,z1)  lie in this domain, then

PQuds=u(x1,y1,z1)-u(x0,y0,z0) (1)

where the line integralMathworldPlanetmath of the left hand side is taken along an arbitrary path in D.

The equation (1) is illustrated by the fact that


is the total differentialMathworldPlanetmath of u, and thus

Title gradient theorem
Canonical name GradientTheorem
Date of creation 2013-03-22 19:11:16
Last modified on 2013-03-22 19:11:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 26B12
Synonym fundamental theorem of line integrals
Related topic LaminarField
Related topic Gradient