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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 Φ = − ≤ ≤ ≤ ≤
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 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 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
Background image of page 2
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)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/24/2011 for the course MATH 200 taught by Professor Jamesdcampbell during the Spring '10 term at Santa Barbara City.

Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online