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The length of the arc will be the least upper bound of the arc lengths of its segments. We shall show that arc length so introduced possesses the usual namely: 1. If the segment A'B' of the curve y is a subset of the segment AB and if the segment AB is rectifiable, then the segment A'B' is also rectifiable and the length of its arcs(A'B') is less than the properties, arc length s(AB) of the segment AB. 2. If C is a point on the segment distinct rectifiable, AB of the curve y which and B, and the segments AC and then the segment AB is also rectifiable, and from both A s(AC) + s(CB) PROOF.

If n is sufficiently large, each of these segments permits a smooth parametrization. In fact, let us assume the contrary. " Suppose a segment t n 't n can be found for every n which does not permit a smooth parametrization. The sequence of segments tn'tn" contains a subsequence of segments whose endpoints t n and in' converge, obviously to a common limit t$. But the point to has a neighborhood which permits a smooth parametrization. For suf" lies in this neighborhood and, ficiently large n the segment t n 't n consequently, it permits a smooth parametrization.

Projected onto the by means of parallel straight lines which form an angle z-axis. Find the equation of the projection. For what x, y-plane $ with the ft will the projection have singular points? Discuss the nature of the singular points. ANSWER: If the projecting lines are parallel to the the equations of the projection will be % a cos co/, y ct tan ft + sin y, 2-plane, ojt. then CHAPTER 8 I, The projection 19 will have singular points if tan & = aa)/c. The singu- lar points are turning points of the first kind.

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Complex manifolds and Hermitian differential geometry by Hwang A.D.

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