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 Rn 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.


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

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