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Unformatted text preview: 16.6: PARAMETRIC SURFACES AND THEIR AREAS KIAM HEONG KWA 1. Parametric Surfaces Let (1.1) r ( u,v ) = x ( u,v ) i + y ( u,v ) j + z ( u,v ) k be a vectorvalued function of two variables on a plane region D in the uvplane. The collection of all points ( x,y,z ) R 3 such that (1.2) x = x ( u,v ) , y = y ( u,v ) , z = z ( u,v ) , ( u,v ) D, i.e., (1.3) S = { ( x ( u,u ) ,y ( u,v ) ,z ( u,v )) R 3  ( u,v ) D } , is called a parametric surface . Equations (1.1) and (1.2) are respec tively called the vector equation and the parametric equations of S . The variables u and v are usually referred to as parameters. The sur face S is traced out by the tip of the position vector r ( u,v ) as ( u,v ) varies throughout the parameter domain D . Example 1 (Problem 3 in the text) . The vector equation r ( u,v ) = ( u + v ) i + (3 v ) j + (1 + 4 u + 5 v ) k determines a plane in R 3 . To see this, write the vector equation in its equivalent parametric equations x = u + v, y = 3 v, z = 1 + 4 u + 5 v. Solving for u and v from the first two parametric equations yields u = x + y 3 , v = 3 y. Substituting for u and v in terms of x and y in the last parametric equation gives z = 1 + 4( x + y 3) + 5(3 v ) or 4 x y z = 4 , which is an equation of a plane. In general, a vector equation of the form r ( u,v ) = r + u a + v b , Date : November 17, 2010. 1 2 KIAM HEONG KWA where r = h x ,y ,z i , a = h a 1 ,a 2 ,a 3 i , and b = h b 1 ,b 2 ,b 3 i are vectors in R 3 such that (1) a 6 = , (2) b 6 = , and (3) a and b are nonparallel (i.e., a b 6 = ) represents a plane in R 3 that contains the point ( x ,y ,z ) and has a b as a normal vector. To see this, write the vector equation as r ( u,v ) r = u a + v b . Then ( r ( u,v ) r ) a b = u a a b + v b a b = 0 because a a b = b a b = . Let us apply the general statement to the vector equation r ( u,v ) = ( u + v ) i + (3 v ) j + (1 + 4 u + 5 v ) k = 3 j + k + u ( i + 4 k ) + v ( i j + 5 k ) . Identifying the vectors r , a , and b , one has r = 3 j + k , a = i + 4 k , and b = i j + 5 k . In particular, a b = 4 i j k . Denoting r ( u,v ) = x i + y j + z k , one has [ x i + ( y 3) j + ( z 1) k ] (4 i j k ) = 4 x y z + 4 = 0 . Example 2 (Problem 6 in the text) . The vector equation r ( s,t ) = h s sin2 t,s 2 ,s cos2 t i defines a circular paraboloid whose axis is the y axis. To see this, consider the equivalent parametric equations x = s sin2 t, y = s 2 , z = s cos2 t. Note that x 2 + z 2 = s 2 (sin 2 2 t + cos 2 2 t ) = s 2 = y. Traces parallel to the xzplanes are circles, while traces parallel to the xy and yzplanes are parabolas....
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This note was uploaded on 11/11/2011 for the course MATH 254.01 taught by Professor Kwa during the Fall '10 term at Ohio State.
 Fall '10
 Kwa
 Calculus, Geometry

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