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16-4 - Math 200 Spring 2010 Handout#26 Section 16-4...

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Math 200, Spring 2010 Handout #26 Section 16-4 Parametrized Surfaces and Surface Integrals Parametrized Surfaces A parametric surface S is the set of all points ( ) , , x y z in 3 described by the function ( ) ( ) ( ) ( ) ( ) , , , , , , u v x u v y u v z u v Φ = where u and v (called parameters) vary throughout parameter domain D in the uv -plane. 1. Identify the surface with parametrization ( ) ( ) 2 , , sin2 , cos2 u v u v u u v Φ = . Grid Curves Suppose a parametrized surface S is given by function ( ) ( ) ( ) ( ) ( ) , , , , , , u v x u v y u v z u v Φ = . If v is kept constant by putting 0 v v = , then ( ) 0 , u v Φ defines a curve 1 C lying on S . Similarly, if u is kept constant by putting 0 u u = , then ( ) 0 , u v Φ defines a curve 2 C lying on S . These curves are called grid curves . 2. Use Winplot to graph the parametric surfaces and indicate on the graph which grid curves have u constant and which have v constant. (a) ( ) ( ) 2 , , sin2 , cos2 , 0 2, 0 2 u v u v u u v u v π Φ = (b) ( ) ( ) , , cos , sin , 5 5, 0 2 u v u v u v v u v π Φ =
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Standard Parametrizations A parametrization for a cylinder of radius R with equation 2 2 2 x y R + = : (using cylindrical coordinates) ( ) ( ) , cos , sin , , 0 2 , z R R z z θ θ θ θ π Φ = −∞ ≤ ≤ ∞ = = n n A parametrization for a sphere of radius R : (using spherical coordinates) ( ) ( ) , sin cos , sin sin , cos ), 0 , 0 2 R R R φ θ φ θ φ θ φ φ π θ π Φ = = = n n A parametrization for a surface given as the graph of a function ( ) , z f x y = : ( ) ( ) ( ) , , , , x y x y f x y Φ = , where x and y parameters = = n n
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3. Find a parametric representation for the surface: (a) the part of the sphere 2 2 2 16 x y z + + = that lies between the planes 2 z = − and 2 z = . (b) the part of the circular paraboloid 2 2 4 x y z + + = that lies in front of the plane 0 x = . (c) the part of the plane 3 z x = + that lies inside the cylinder 2 2 1 x y + = . glyph1197ormal Vectors and Tangent Planes Suppose a parametrized surface S is given by function ( ) ( ) ( ) ( ) ( ) , , , , , , u v x u v y u v z u v Φ = and point P is a point on this surface. Then, at point P , the tangent vector to grid curve 1 C defined by ( ) 0 , u v Φ [ v is held constant] is ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 , , , , , ,
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