Definition:Conservative Vector Field

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Definition

Let $R$ be a region of space.

Let $\mathbf V$ be a vector field acting over $R$.


Definition 1

$\mathbf V$ is a conservative vector field if and only if the contour integral over $\mathbf V$ around every simple closed contour is zero:

$\ds \oint \mathbf V \cdot \d \mathbf l = 0$


Definition 2

$\mathbf V$ is a conservative vector field if and only if its curl is everywhere zero:

$\curl \mathbf V = \bszero$


Also known as

A conservative vector field is also known in the literature as:

a conservative field
a scalar potential field (from its property that it is the gradient of some scalar field)
a non-curl field
an irrotational field
a lamellar field.


A vector in such a conservative vector field is sometimes known as a lamellar vector.


Examples

Electric Field

Let $F$ be an electric potential field over a region of space $R$.

Let $F$ give rise to the electric field $\mathbf V$.

Then $\mathbf V$ is a conservative vector field, as it is the gradient of $F$.


Also see

  • Results about conservative vector fields can be found here.


Linguistic Note

The adjective lamellar derives from the Latin noun lamella, which means thin layer.

The lamellae to which lamellar field refers are the equal surfaces of the scalar field from which the lamellar vector field is given rise to by way of the gradient operator.


Sources