Curves and Differential Forms

In this chapter we discuss notions such as force, work, vector field, differential form, conservative vector field and its potential, and the solvability in an open set Ω R_n of the equation

$${\rm grad} \, U = F$$

We shall see that the vector field F is conservative, i.e., the equation grad U = F is solvable, if and only if the work along closed curves in Ω is zero, and we shall discuss how to compute a solution, a potential.

When n = 3, every function U of class C² satisfies the equation rot grad U = 0. Therefore, rot F = 0 in Ω is a necessary condition in order for the vector field F ∈ C¹ to be conservative in Ω. In terms of differential forms, we shall also see that rot F = 0 suffices for F to be conservative in simply connected domains.

Though Lebesgue’s theory of integration would allow us more general results, here we prefer to limit ourselves to the use of Riemann integral.