Closed and Exact Differential Forms - Examples

Examples

The simplest example of a form which is closed but not exact is the 1-form "dθ" (quotes because it is not the derivative of a globally defined function), defined on the punctured plane which is locally given as the derivative of the argument - note that argument is locally but not globally defined, since a loop around the origin increases (or decreases, depending on direction) the argument by 2π, which corresponds to the integral:

and for general paths is known as the winding number. The differential of the argument is however globally defined (except at the origin), since differentiation only requires local data and different values of the argument differ by a constant, so the derivatives of different local definitions are equal; this line of thought is generalized in the notion of covering spaces.

Explicitly, the form is given as:

which is not defined at the origin. This can be computed from a formula for the argument, most simply via arctan(y/x) (y/x is the slope of the line passing through (x,y), and arctan converts slope to angle), recognizing 1/(x2+y2) as corresponding to the derivative of arctan, which is 1/(x2+1) (these agree on the line y=1). While the differential is correctly computed by symbolically differentiating this expression, this formula is only strictly correct on the halfplane x>0, and properly one must use a correct formula for the argument.

This form generates the de Rham cohomology group meaning that any closed form is the sum of an exact form and a multiple of where accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a potential function) being the derivative of a globally defined function.