14.5 Notes (Parametric Surfaces)

14.5 Notes (Parametric Surfaces) - v u r r is a vector...

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Calculus III – Section 14.5 – Parametric Surfaces Since you are now pros at parametrizing curves, now you will learn how to represent a surface by a set of parametric equations – or by a vector-valued function. For a curve you needed one parameter (usually t) so, of course, for a surface you will need two parameters (usually u and v). DEFINITION Let x, y, and z be functions of u and v that are continuous on a domain D in the uv-plane. The set of points (x,y,z) given by r (u,v) = x(u,v) I +y(u,v) j + z(u,v) k is called a parametric surface . The equations x=x(u,v), y=y(u,v), and z=z(u,v) are the parametric equations for the surface. Example: Identify and sketch the surface represented by k u j v u i v u v u r + + = sin cos ) , ( . Example: Find a vector-valued function whose graph is the cylinder 16 4 2 2 = + y x Note that the vectors u r and v r are both tangent to the surface so
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Unformatted text preview: v u r r is a vector normal to the surface. If v u r r for any (u,v) in D, the surface is called smooth and will have a tangent plane. In other words, the surface will have no sharp points or cusps. Area of a Parametric Surface Let S be a smooth parametric surface r (u,v) = x(u,v) I +y(u,v) j + z(u,v) k defined over an open region D in the uv-plane. If each point on the surface S corresponds to exactly one point in the domain D, the surface area of S is given by Surface area = = D v u S dA r r dS Where k u z j u y i u x r u + + = and k v z j v y i v x r v + + = . Example: Find the area of the part of the paraboloid k u j v u i v u v u r 2 sin 2 cos 2 ) , ( + + = where 2 u and 2 v ....
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This note was uploaded on 01/13/2012 for the course MATH 2574 taught by Professor Pamelasatterfield during the Spring '12 term at NorthWest Arkansas Community College.

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14.5 Notes (Parametric Surfaces) - v u r r is a vector...

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