Definition:Conservative Vector Field

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 line 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 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

Electrostatic Field

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

Let $\mathbf V$ be the electric field to which $F$ gives rise to.

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

Also see

• Equivalence of Definitions of Conservative Vector Field

• Vector Field is Expressible as Gradient of Scalar Field iff Conservative

• 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.