In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two points: it is path independent. Conversely, path independence is equivalent to the vector field being conservative. Conservative vector fields are also irrotational, meaning that (in three-dimensions) they have vanishing curl. In fact, an irrotational vector field is necessarily conservative provided that a certain condition on the geometry of the domain holds: it must be simply 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.
Read more about Conservative Vector Field: Definition, Path Independence, Irrotational Vector Fields, Irrotational Flows, Conservative Forces
Famous quotes containing the words conservative and/or field:
“Almost always tradition is nothing but a record and a machine-made imitation of the habits that our ancestors created. The average conservative is a slave to the most incidental and trivial part of his forefathers’ glory—to the archaic formula which happened to express their genius or the eighteenth-century contrivance by which for a time it was served.”
—Walter Lippmann (1889–1974)
“Mine was, as it were, the connecting link between wild and cultivated fields; as some states are civilized, and others half-civilized, and others savage or barbarous, so my field was, though not in a bad sense, a half-cultivated field. They were beans cheerfully returning to their wild and primitive state that I cultivated, and my hoe played the Ranz des Vaches for them.”
—Henry David Thoreau (1817–1862)