Definition:Conservative Vector Field/Definition 1

Definition

Let $R$ be a region of space.

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

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

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

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:

an irrotational vector

a lamellar vector.

Also see

• Equivalence of Definitions of Conservative Vector Field

• 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

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $2 a$. The Operation $\nabla S$