Conservative vector field

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转自：http://en.wikipedia.org/wiki/Conservative_vector_field

In vector calculus aconservative vector field is avector field which is thegradient of afunction, known in this context as ascalar potential. Conservative vector fields have the property that theline integral from one point to another is independent of the choice of path connecting the two points: it ispath independent. Conversely, path independence is equivalent to the vector field being conservative. Conservative vector fields are alsoirrotational, meaning that (in three-dimensions) they have vanishingcurl. In fact, an irrotational vector field is necessarily conservative provided that a certain condition on the geometry of the domain holds: it must besimply connected.

An irrotational vector field which is also solenoidal is called a Laplacian vector field because it is the gradient of a solution of Laplace's equation.

Path independence

A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that $S\subseteq\mathbb{R}^3$ is a region of three-dimensional space, and that $P$ is a rectifiable path in $S$ with start point $A$ and end point $B$ . If $\mathbf{v}=\nabla\varphi$ is a conservative vector field then thegradient theorem states that

$\int_P \mathbf{v}\cdot d\mathbf{r}=\varphi(B)-\varphi(A).$

This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.

An equivalent formulation of this is to say that

$\oint \mathbf{v}\cdot d\mathbf{r}=0$

for every closed loop in S. The converse is also true: if the circulation of v around every closed loop in an open set S is zero, then v is a conservative vector field.