Examples in Low Dimensions
Differential forms in R2 and R3 were well known in the mathematical physics of the nineteenth century. In the plane, 0-forms are just functions, and 2-forms are functions times the basic area element dx∧dy, so that it is the 1-forms
that are of real interest. The formula for the exterior derivative d here is
where the subscripts denote partial derivatives. Therefore the condition for α to be closed is
In this case if h(x,y) is a function then
The implication from 'exact' to 'closed' is then a consequence of the symmetry of second derivatives, with respect to x and y.
Read more about this topic: Closed And Exact Differential Forms
Famous quotes containing the words examples and/or dimensions:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)