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Unformatted text preview: Primer on Surface Integrals Overview Two dimensional surfaces can be represented by a scalar equation of the form (1) g H x, y, z L = or z = f H x, y L It is convenient to parameterize the surface in terms of two parameters. A position vector the defines a point of the surface can be represented as (2) r H u, v L = x 1 H u, v L e 1 + x 2 H u, v L e 2 + x 3 H u, v L e 3 For a smooth surface given explicitly in the form z = f H x ; y L , a point (x; y; z) is said to be above the surface if z > f H x ; y L and below the surface if z < f H x ; y L . Hence we can define two different sides of the surface, and therefore at each point of a surface S there are potentially two normals. The existence of two normals depends on whether or not the surface is orientable. A surface S is called orientable if a unit normal vector n can be defined at every nonboundary point of S in such a way that the set of normal vectors varies continuously over all S. In particular, consider any smooth closed curve ! on S, passing through a point P, Let M H u , v L be any point on ! . Then at each point M choose that direction of the normal so that it varies continuously from P along the curve ! . As the curve returns to P the normal may either return to be that chosen at P or be the opposite one. If the normal is of the same sign, then S is called an oriented surface and has two distinct sides. A Möbius strip is an example of a surface that is not orientable and is onesided. The position vector for a point on the Möbius surface is (3) r H u, v L = H R + u Cos @ v ê 2 DL Cos @ v D e 1 + H R + u Cos @ v ê 2 DL Sin @ v D e 2 + u Sin @ v ê 2 D e 3 Shown below is a graph of a Möbius surface (taken from Mathworld) which illustrates the onesideness of the surface. Figure 1 Consider now an orientable surface S that is described in terms of the parameters u and v with parametric equa tions of the type: (4) x 1 = x 1 H u, v L , x 2 = x 2 H u, v L , x 3 = x 3 H u, v L Figure 2 is a schematic of a surface patch taken from S Figure 2 P u = C 1 u = C 2 v = K 1 v = K 2 r ÅÅÅÅÅÅÅ v r ÅÅÅÅÅÅÅ u n The position vector that defines a point P on the surface S is given by (5) r H u, v L = x 1 H u, v L e 1 + x 2 H u, v L e 2 + x 3 H u, v L e 3 Thus for fixed u, say u = C 1 , the position vector r H C 1 , v L traces out a coordinate curve v on the surface. Similarly, for v = K 1 , the position vector r H u, K 1 L traces out a coordinate curve u on the surface. It follows then that a tangent vector to the coordinate curve v = K 1 is (6) r ÅÅÅÅÅÅÅ u = x 1 ÅÅÅÅÅÅÅÅÅ u e 1 + x 2 ÅÅÅÅÅÅÅÅÅ u e 2 + x 3 ÅÅÅÅÅÅÅÅÅ u e 3 and similarly, the tangent vector to the curve u = C 1 is (7) r ÅÅÅÅÅÅÅ v = x 1 ÅÅÅÅÅÅÅÅÅ v e 1 + x 2 ÅÅÅÅÅÅÅÅÅ v e 2 + x 3 ÅÅÅÅÅÅÅÅÅ v e 3 We will assume that these derivatives exist, are continuous and non parallel everywhere on the surface. FromWe will assume that these derivatives exist, are continuous and non parallel everywhere on the surface....
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 Spring '10
 Franics
 Vector Calculus, Flux, Surface, Orientability, Möbius surface

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