Definition:Conservative Vector Field
Definition
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 scalar potential field (from its property that it is the gradient of some scalar field)
A vector in such a conservative vector field is sometimes known as:
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conservative field
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conservative field