Conservative vector field

A vector field that is the gradient of a scalar potential. Its line integrals depend only on endpoints, so every closed-curve integral is zero.

Gradient theorem for line integrals: A theorem stating that a gradient field’s line integral equals the potential’s change between the endpoints. It directly converts the field’s path integral into a difference of potential values.
Vector field: A function assigning a vector to each point in a region of space. Conservativeness is a special property of vector fields.
Potential energy: Stored energy associated with an object's position or configuration in a conservative force field. A conservative force can be represented as the negative gradient of potential energy.
Nonconservative force: A force whose work depends on the path between two positions. Unlike conservative forces, it cannot be represented globally by a single-valued potential energy.
Potential function: A scalar-valued function whose gradient produces a specified vector field. Its gradient is the defining source of a conservative vector field.
Gradient: A vector of a scalar function’s partial derivatives, pointing toward its direction of greatest increase. A field is conservative precisely when it equals a scalar function’s gradient.
Gravitational potential: Potential energy per unit mass associated with a gravitational field. Its spatial gradient, with a minus sign, gives gravitational acceleration.
Rotational vector field: A vector field with nonzero curl in at least part of its domain. Nonzero curl rules out a local scalar potential in the affected region.
Path independence: A property in which a line integral between fixed endpoints has the same value along every path. It is equivalent to conservativeness on a path-connected domain.
Line integral: An integral that accumulates a scalar or vector quantity along a curve. Path independence concerns line integrals of the field along curves.

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