chapter10 - Chapter 10. Surface Integrals 10.1 Parametric...

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Unformatted text preview: Chapter 10. Surface Integrals 10.1 Parametric Surfaces A parametric representation of a surface is given by the two-variable vector function r ( u,v ) = x ( u,v ) i + y ( u,v ) j + z ( u,v ) k (1) where u and v are two independent parameters. The collection of points with position vectors (1) form a surface in the xyz-space. The equations x = x ( u,v ) ,y = y ( u,v ) ,z = z ( u,v ) are called the parametric equations of the sur- face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . ...... x ......................................................................... . . . . . . . . . . . y .................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. .. . .. .. . .. .. .. . .. .. . • O r ( u,v ) 2 MA1505 Chapter 10. Surface Integrals 10.1.1 Example (Planes) For a general plane ax + by + cz = d , we can let two of the three components be u and v and obtain the remaining component in terms of u and v using the above equation. E.g. 3 x +2 y- 4 z = 6: Let x ( u,v ) = u, y ( u,v ) = v . Then z ( u,v ) = 1 4 (3 x + 2 y- 6). So the parametric representation of this plane is r ( u,v ) = u i + v j + 1 4 (3 u + 2 v- 6) ¶ k . If one variable is absent from the equation, we let the missing component be u or v . E.g. 2 y + x = 7: Let z ( u,v ) = u . Then y ( u,v ) = v 2 3 MA1505 Chapter 10. Surface Integrals and x ( u,v ) = 7- 2 v . r ( u,v ) = (7- 2 v ) i + v j + u k . If two variables are absent from the equation, we let the two missing components be u or v . E.g. The xy-plane is given by r ( u,v ) = u i + v j + 0 k . 10.1.2 Example (Surfaces of the form z = f ( x,y ) ) A natural parametric representation of S is r ( u,v ) = u i + v j + f ( u,v ) k E.g. The paraboloid z = x 2 + y 2 . r ( u,v ) = u i + v j + ( u 2 + v 2 ) k . 3 4 MA1505 Chapter 10. Surface Integrals E.g. The upper cone z = p x 2 + y 2 . r ( u,v ) = u i + v j + √ u 2 + v 2 k . 10.1.3 Example (Spheres ) We have a standard parametric representation for a sphere x 2 + y 2 + z 2 = a 2 of radius a centered at the origin: r ( u,v ) = ( a sin u cos v ) i +( a sin u sin v ) j +( a cos u ) k . E.g. When 0 ≤ u ≤ π, ≤ v ≤ 2 π , the represen- tation gives the full sphere. When 0 ≤ u ≤ π/ 2 , ≤ v ≤ 2 π , the representa- tion gives the upper hemisphere. 4 5 MA1505 Chapter 10. Surface Integrals 10.1.4 Example (Circular cylinders) We have a standard parametric representation for cir- cular cylinder x 2 + y 2 = a 2 about the z-axis: r ( u,v ) = ( a cos u ) i + ( a sin u ) j + v k ....
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chapter10 - Chapter 10. Surface Integrals 10.1 Parametric...

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