Some Remarks Concerning Integrals of Curvature on Curves and Surfaces

In this paper we discuss some topics that came up in Chapters 2 and 3 of Part III of [8]. These involve relations between derivatives of Cauchy integrals on curves and surfaces and curvatures of the curves and surfaces. In R for n > 2, “Cauchy integrals” can be based on generalizations of complex analysis using quarternions or Clifford algebras (as in [3]). Part of the point here is to bring out the basic features and types of computations in a simple way, if not finer aspects which can also be considered. Let us consider first curves in the plane R. We shall identify R with the set C of complex numbers. Let Γ be some kind of curve in C, or perhaps union of pieces of curves. For each z ∈ C\Γ, we have the contour integral