Pages similar to: The curl of a gradient is zero
- The idea of the curl of a vector field
Intuitive introduction to the curl of a vector field. Interactive graphics illustrate basic concepts. - Subtleties about curl
Counterexamples illustrating how the curl of a vector field may differ from the intuitive appearance of a vector field's circulation. - The components of the curl
Illustration of the meaning behind the components of the curl. - Divergence and curl notation
Different ways to denote divergence and curl. - Divergence and curl example
An example problem of calculating the divergence and curl of a vector field. - 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. - 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 definition of curl from line integrals
How the curl of a vector field is defined by line integrals representing circulation. - 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. - The gradient theorem for line integrals
A introduction to the gradient theorem for conservative or path-independent 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. - 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. - 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. - Proper orientation for Stokes' theorem
The importance of orientating the surface and its boundary correctly when using Stokes' theorem. - Stokes' theorem examples
Examples illustrating how to use Stokes' theorem. - The gradient vector
The gradient vector is the matrix of partial derivatives of a scalar valued function viewed as a vector.
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