Vector Analysis pp 65-78 | Cite as

# The Concept of Orientation

Chapter

## Abstract

As you know, the
The
where

*direction*of integration matters when you integrate a function of a real variable:$$\int\limits_a^b {f(x)dx = - } \int\limits_b^a {f(x)dx.} $$

*dx*senses, so to speak, when the direction of integration is reversed: the differences Δ*x*_{ k }=*x*_{ k }_{+1}−*x*_{ k }in the Riemann sums Σ*f*(*x*_{ k })Δ*x*_{ k }are positive or negative according to whether the partition points are increasing or decreasing. The same thing happens with line integrals$$\int\limits_\gamma {f(x,y,z)dx + g(x,y,z)dy + h(x,y,z)dz,} $$

*γ*is a curve in ℝ^{3}, 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.## Keywords

Vector Space Tangent Space Real Vector Space Path Component Local Coherence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 2001