Gradient theorem

The gradient theorem states that the line integral of a scalar field’s gradient along a path equals the field’s value at the endpoint minus its value at the starting point.

Line integral: An integral that accumulates a scalar or vector quantity along a curve. The theorem evaluates this integral when the vector field is a gradient.
Scalar field: A function assigning a scalar value to each point in a domain. The theorem measures how this function changes between the path’s endpoints.
Potential energy: A scalar quantity associated with a system’s position in a conservative force field. The theorem computes work from changes in a potential, with sign determined by the force convention.
Stokes' theorem: A theorem relating a surface integral of curl to a line integral around its boundary. It concerns circulation around a closed boundary, not endpoint changes along a path.
Gradient: A vector field pointing in the direction of greatest increase of a scalar field, with magnitude equal to its rate of increase. Integrating this field along a path tracks the scalar field’s change.
Vector field: A function assigning a vector to each point in a domain. The gradient of the scalar field is the vector field integrated along the path.
Work (physics): Energy transferred when a force acts through a displacement. For a conservative force, work along a path is determined by endpoint potential values.
Divergence theorem: A theorem relating flux through a closed surface to divergence integrated over its interior. It connects flux and volume rather than a gradient integral and endpoint values.
Chain rule: A differentiation rule for a function composed with another function. Applied to the scalar field along a parametrized path, it turns the integrand into a derivative along the path.
Parametrized curve: A curve described by a position-valued function of a parameter. Parametrizing the path lets the chain rule express the line integral as an ordinary integral.

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Nonconservative forceRelated: It explains why conservative forces have path-independent work.