gradient theorem

If  $u=u(x,y,z)$  is continuously differentiable function in a simply connected domain (http://planetmath.org/Domain2) $D$ of ${ℝ}^3$ and  $P=(x_0,y_0,z_0)$  and  $Q=(x_1,y_1,z_1)$  lie in this domain, then

${\displaystyle {\int }_P^Q}\nabla u\cdot \overrightarrow{ds}=u(x_1,y_1,z_1)-u(x_0,y_0,z_0)$ (1)

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

The equation (1) is illustrated by the fact that

$$\nabla u\cdot \overrightarrow{ds}=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz$$

is the total differential of $u$, and thus

$${\int }_P^Q\nabla u\cdot \overrightarrow{ds}={\int }_P^Q𝑑u.$$

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