Conservative vector field ↩

shorthands: {}
aliases: [conservative field, Conservative field, gradient field, Gradient field, Irrational vector field]
created: 2021-10-19 20:15:35
modified: 2022-06-24 00:18:19

A conservative vector field is a vector field that is the gradient of some scalar field.

Other words: is a conservative vector field if such that . Then is called the scalar potential for .

Relation with curl: if a vector field's curl vanishes everywhere, then it is conservative.

Path independence

A key property: an integral along a path does not depend on the route taken, only on the endpoints of that path.
Then the gradient theorem states that:

Or:

Where is any closed path in the domain of .