The Concept of Orientation

Abstract

As you know, the direction of integration matters when you integrate a function of a real variable:

$$\int\limits_a^b {f(x)dx = - } \int\limits_b^a {f(x)dx.} $$

The dx senses, so to speak, when the direction of integration is reversed: the differences Δx _k = x _{k +1} − x _k in the Riemann sums Σf (x _k )Δx _k are positive or negative according to whether the partition points are increasing or decreasing. The same thing happens with line integrals

$$\int\limits_\gamma {f(x,y,z)dx + g(x,y,z)dy + h(x,y,z)dz,} $$

where γ is a curve in ℝ³, and with contour integrals ∫ _γ f (z) dz in complex function theory. They are invariant under all reparametrizations of the curve that do not change the direction in which the curve is traced. But if the curve is traced backwards, the sign of the integral is reversed.