If  $u=u(x,\,y,\,z)$  is continuously differentiable function in a simply connected domain (http://planetmath.org/Domain2) $D$ of $\mathbb{R}^{3}$ and  $P=(x_{0},\,y_{0},\,z_{0})$  and  $Q=(x_{1},\,y_{1},\,z_{1})$  lie in this domain, then

 $\displaystyle\int_{P}^{Q}\!\nabla u\!\cdot\!\vec{ds}\;=\;u(x_{1},\,y_{1},\,z_{% 1})-u(x_{0},\,y_{0},\,z_{0})$ (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

 $\nabla u\!\cdot\!\vec{ds}\;=\;\frac{\partial u}{\partial x}dx+\frac{\partial u% }{\partial y}dy+\frac{\partial u}{\partial z}dz$

is the total differential of $u$, and thus

 $\int_{P}^{Q}\!\nabla u\!\cdot\!\vec{ds}\;=\;\int_{P}^{Q}\!du.$
Title gradient theorem GradientTheorem 2013-03-22 19:11:16 2013-03-22 19:11:16 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 26B12 fundamental theorem of line integrals LaminarField Gradient