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13-06-flux-integrals

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

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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.

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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. 5 Evaluate the surface integral RR S F · d S , where F = h- z, x, y i and S is the full unit hemisphere (including the base!) on and above the xy -plane (so x 2 + y 2 + z 2 = 1 plus a disk) with the outward orientation.
Flux Integrals – Answers and Solutions 1 Here we use the parameterization r ( x, y ) = h x, y, 1 - 1 2 x - 1 2 y i . From this we find that r x × r y = i j k 1 0 - 1 2 0 1 - 1 2 = 1 2 , 1 2 , 1 . Therefore ZZ S (1 + z ) dS = ZZ D 1 + 1 - 1 2 x - 1 2 y dA. The region D in our parameter space (the xy -plane) need to cover our surface is the triangle D = { ( x, y ) : x 0 , y 0 , x + y 2 } , so we can write our limits as follows: ZZ S (1 + z ) dS = Z 2 0 Z 2 - x 0 2 - 1 2 x - 1 2 y dy dx = Z 2 0 1 4 x 2 - 2 x + 3 dx = 1 12 · 2 3 - 2 2 + 3 · 2 = 8 3 .

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

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