Pages similar to: How to determine if a vector field is conservative
- The gradient theorem for line integrals
A introduction to the gradient theorem for conservative or path-independent line integrals. - An introduction to conservative vector fields
An introduction to the concept of path-independent or conservative vector fields, illustrated by interactive graphics. - 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. - A conservative vector field has no circulation
How a conservative, or path-independent, vector field will have no circulation around any closed curve. - Finding a potential function for conservative vector fields
How to find a potential function for a given conservative, or path-independent, vector field. - 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 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. - Testing if three-dimensional vector fields are conservative
Examples of testing whether or not three-dimensional vector fields are conservative (or path-independent). - Vector field overview
An overview introducing the basic concept of vector fields in two or three dimensions. - Vector fields as fluid flow
Interpretation of vector fields as velocity fields of fluids. - 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. - Directional derivative and gradient examples
Examples of calculating the directional derivative and the gradient. - Derivation of the directional derivative and the gradient
Derivation of the directional derivative and the gradient from the definition of differentiability of scalar-valued multivariable functions. - 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. - Alternate notation for vector line integrals
An alternative notation for the line integral of a vector field is introduced. - Line integrals as circulation
Definition of circulation as the line integral of a vector field around a closed curve. - Vector line integral examples
Example of calculating line integrals of vector fields. - 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 curl of a gradient is zero
Calculation showing that the curl of a gradient is zero. - The fundamental theorems of vector calculus
A summary of the four fundamental theorems of vector calculus and how the link different integrals.