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# gradient theorem

If $u=u(x,\,y,\,z)$ is continuously differentiable function in a simply connected domain $D$ 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

$\int_{P}^{Q}\!\nabla u\!\cdot\!\vec{ds}\;=\;\int_{P}^{Q}\!du.$ |

Related:

LaminarField, Gradient

Synonym:

fundamental theorem of line integrals

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

26B12*no label found*

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