13-06-flux-integrals

13-06-flux-integrals - Math 21a Flux Integrals Spring, 2009...

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Unformatted text preview: Math 21a Flux Integrals Spring, 2009 A surface integral is ZZ S f ( x,y,z ) dS = ZZ D f ( r ( u,v )) | r u r v | du dv, where f is a function defined on the parametric surface r ( u,v ). 1 Evaluate the surface integral ZZ S (1 + z ) dS, where S is that part of the plane x + y + 2 z = 2 in the first octant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ....... ......................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . . . . . . . . y ...... ...... z ...... .......................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................... . . . . . . . . . . . . . . . . . . . Suppose F is a continuous vector field on an oriented surface S with unit normal vector n . The surface integral of F over S is ZZ S F d S = ZZ S F n dS = ZZ D F ( r u r v ) du dv for a parametrically defined surface. 2 Evaluate the surface integral RR S F d S , where F = y i- x j + z k and S is the part of the sphere x 2 + y 2 + z 2 = 4 in the first octant with inward orientation. 3 Evaluate the surface integral RR S F d S , where F = h x,y, 2 z i and S is the part of the paraboloid z = 4- x 2- y 2 that lies above the unit square [0 , 1] [0 , 1] with the downward orientation. 4 Evaluate the surface integral RR S F d S , where F = x i + y j + (2 x + 2 y ) k and S is the part of the paraboloid z = 4- x 2- y 2 that lies above the unit disk (centered at the origin) with upward orientation....
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This note was uploaded on 11/15/2011 for the course CS 50 taught by Professor Malan during the Spring '08 term at Harvard.

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13-06-flux-integrals - Math 21a Flux Integrals Spring, 2009...

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