Examining Oriented Curvilinear Integrals

Abstract

In the present chapter, the question of the so-called “oriented integrals” over surfaces is undertaken. The first notion to be introduced is that of the orientation. It will be explained in detail when solving particular problems and the formal definition is as follows. Given a linear k-dimensional space over $\mathbb R$ and its basis e = (e ₁, e ₂, …, e _k). The elements of any other basis f = (f ₁, f ₂, …, f _k) are linear combinations of e _i, which can be written in the matrix form

$$\displaystyle f=Me. $$

The matrix M is quadratic and nonsingular, so either $\det M>0$ or $\det M<0$.