By Felix Klein

This quantity introduces Riemann's method of multiple-value services and the geometrical illustration of those features via Riemann surfaces. It concentrates at the kind of services that may be outlined upon those surfaces, limiting the remedy to rational features and their integrals demonstrating how Riemann's mathematical rules approximately Abelian integrals should be derived by means of reflecting upon the move of electrical present on surfaces. This paintings is without doubt one of the top introductions to the origins of topological difficulties.

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1867, pp. 169 et seq. ] introductory remarks. 22 be particularly well observed in Plateau’s experiments. We shall attempt to define such states of motion also by a potential and we shall especially enquire what is the case in steady motion. The proper extension of our conception of a potential presents itself at once. Let u be a function of position on the surface and let the curves u = const. be drawn; moreover let the direction of fluid-motion on the surface at every point be perpendicular to the curve u = const.

It is more difficult to prove the converse, that the equality of the p’s is a sufficient condition for the possibility of a uniform correspondence between the two surfaces. For proof of this ∗ Deformations by means of continuous functions only are considered here. Moreover in the arbitrary surfaces of the text certain particular occurrences are for the present excluded. It is best to imagine them without singular points; branch-points and hence the penetration of one sheet by another will be considered later on (§ 13).

Riemann’s theory. 45 In the conjugate streaming, the curves of latitude play the part of the meridians in the first example; this is shown in the following drawing: The direction of motion in this case is the same on the upper and lower sides. Let us now deform the anchor-ring, p = 1, by causing two excrescences to the right of the figure, roughly speaking, to grow from it, which gradually bend towards each other and finally coalesce. We then have a surface p = 2 and on it a pair of conjugate streamings as illustrated by Figures 23 and 24.