gradient theorem


If  u=u(x,y,z)  is continuously differentiable function in a simply connected domain (http://planetmath.org/Domain2) 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

uds=uxdx+uydy+uzdz

is the total differentialMathworldPlanetmath of u, and thus

PQuds=PQ𝑑u.
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