Slide_20 - MA1104 Multivariable Calculus Lecture 20 Dr KU...

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MA1104 Multivariable Calculus Lecture 20 Dr KU Cheng Yeaw Thursday April 2, 2009
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Recap Last lecture ... we defined the surface area of a general parametric surface we introduced surface integral of scalar field over a smooth curve. The main idea is to use parametrization r ( u, v ) for the smooth curve to evaluate such integrals.
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Overview In this lecture ... we define surface integrals of vector fields. We cannot do it for any surface. We deal with only orientable surfaces. The surface integral of a vector field (also called flux) is related to the surface integral of some scalar field. Therefore, we can evaluate these integrals using parametrization. We shall see its physical interpretation. This leads us naturally to the Divergence Theorem.
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Oriented Surface To define surface integrals of vector fields, we need to rule out nonorientable surfaces. What do we mean by oriented surface? Oriented Surface A surface S is orientable (or two-sided ) if it is possible to define a unit normal vector n at each point ( x, y, z ) not on the boundary of the surface and n is a continuous function of ( x, y, z ) .
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An orientable surface S has two identifiable sides (a top and a bottom or an inside and an outside). For example, a sphere is orientable - the two sides are the inside and the outside. You cannot get from the inside to the outside without passing through the sphere. A plane is orientable - you cannot get from one side of the surface to the other so that the unit normal vectors changes continuously!
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All the surface we have seen so far are orientable. But it is possible to construct one nonorientable surface: Taking a long rectangular strip of paper. Giving it a half-twist. Taping the short edges together.
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This famous example is called the obius strip named after the German mathematician A. F. M¨ obius.
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If an ant were to crawl along the M¨ obius strip starting at a point P , it would end up on the ‘other side’ of the stripthat is, with its upper side pointing in the opposite direction.
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Then, if it continued to crawl in the same direction, it would end up back at the same point P without ever having crossed an edge. Therefore, a M¨ obius strip really has only one side.
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From now on, we consider only orientable (two-sided) surfaces. We start with a surface S that has a tangent plane at every point ( x, y, z ) on S (except at any boundary point). There are two unit normal vectors n 1 and n 2 = - n 1 at ( x, y, z ) .
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Since S is orientable, it is possible to choose a unit normal vector n at every such point ( x, y, z ) so that n varies continuously over S . The given choice of n provides S with an orientation.
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There are two possible orientations for any orientable surface.
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How do we find these unit normal vectors? If S is a smooth orientable surface given in parametric form by a vector function r ( u, v ) , then it is automatically supplied with the orientation of the unit normal vector n = r u × r v || r u × r v || .
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