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 We shall see that the vector field

_{n}of the equation$${\rm grad} \, U = F$$

*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 *C*^{2} satisfies the equation rot grad *U* = 0. Therefore, rot *F* = 0 in Ω is a necessary condition in order for the vector field *F ∈ C*^{1} 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.

## Keywords

Closed Curf Area Formula Equation Grad Angle Form Piecewise Smooth Curve
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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009