### Pages similar to: The gradient theorem for line integrals

- How to determine if a vector field is conservative

A discussion of the ways to determine whether or not a vector field is conservative or path-independent. - Finding a potential function for conservative vector fields

How to find a potential function for a given conservative, or path-independent, vector field. - A simple example of using the gradient theorem

An example of using the gradient theorem to calculate the line integral of a conservative, or path-independent, vector field. - A path-dependent vector field with zero curl

A counterexample showing how a vector field could have zero curl but still fail to be conservative or path-independent. - Finding a potential function for three-dimensional conservative vector fields

How to find a potential function for a given three-dimensional conservative, or path-independent, vector field. - A conservative vector field has no circulation

How a conservative, or path-independent, vector field will have no circulation around any closed curve. - An introduction to conservative vector fields

An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics. - The integrals of multivariable calculus

A summary of the integrals of multivariable calculus, including calculation methods and their relationship to the fundamental theorems of vector calculus. - The fundamental theorems of vector calculus

A summary of the four fundamental theorems of vector calculus and how the link different integrals. - The idea behind Green's theorem

Introduction to Green's theorem, based on the intuition of microscopic and macroscopic circulation of a vector field. - The idea behind Stokes' theorem

Introduction to Stokes' theorem, based on the intuition of microscopic and macroscopic circulation of a vector field and illustrated by interactive graphics. - Introduction to a line integral of a vector field

The concepts behind the line integral of a vector field along a curve are illustrated by interactive graphics representing the work done on a magnetic particle. The graphics motivate the formula for the line integral. - Stokes' theorem examples

Examples illustrating how to use Stokes' theorem. - Testing if three-dimensional vector fields are conservative

Examples of testing whether or not three-dimensional vector fields are conservative (or path-independent). - An introduction to the directional derivative and the gradient

Interactive graphics about a mountain range illustrate the concepts behind the directional derivative and the gradient of scalar-valued functions of two variables. - Length, area, and volume factors

A summary of the expansion factors that arise from mappings in integrals of multivariable calculus. - Line integrals are independent of parametrization

Description of how the value of a line integral over a curve is independent of the parametrization of the curve. - Calculating the formula for circulation per unit area

A sketch of the proof for the formula for the component of the curl of a vector field. - Introduction to a line integral of a scalar-valued function

Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve. - The arc length of a parametrized curve

Introduction to the arc length of a parametrized curve. The arc length definition is illustrated with interactive graphics.