vctut06 - , , 0) , (0 , a, 0) and (0 , , a ), taken in that...

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The University of Sydney School of Mathematics and Statistics Tutorial 6 MATH2061: Vector Calculus Summer School 2012 1. Evaluate ii S F · n dS , where F = 4 xz i y 2 j + yz k and S is the surface of the cube bounded by x = 0 , x = 1 , y = 0 , y = 1 , z = 0 and z = 1 . ( n is the unit outward normal to the surface S. ) 2. Find the ±ux of F = x 2 ( y 2 + z 2 ) i + y 2 ( z 2 + x 2 ) j + z 2 ( x 2 + y 2 ) k outwards across the entire surface of the cylindrical region bounded by x 2 + y 2 = 4 , z = 2 and z = 3 . 3. Evaluate ii S ( ∇ × F ) · n dS , where S is the surface of the paraboloid z = 9 x 2 y 2 for z 0, F = (2 x + y ) i + xyz j + 3 k and n is the outwards-pointing unit normal. (Note that there are various ways you could do this. For example, you could cal- culate the integral directly, or use the divergence theorem, or use Stokes’ theorem. You should use whichever method you think will be the easiest.) 4. Evaluate c C y 2 dx + z 2 dy + x 2 dz , where C is the triangle with vertices (0
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Unformatted text preview: , , 0) , (0 , a, 0) and (0 , , a ), taken in that order. Extra questions for outside the tutorial 5. (From the MATH2001 exam, 2000.) Let S be the surface consisting of the hemisphere S 1 de²ned by z = r a 2 − x 2 − y 2 , and the disc S 2 given by x 2 + y 2 ≤ a 2 , z = 0. (a) Find the ±ux of the vector ²eld F = ( ye z + x 3 ) i + ( x 4 sin z + y 3 ) j + z 3 k outwards through the whole surface S . (b) Find also the ±ux of F upwards through the top hemisphere S 1 . (c) Now instead let F be de²ned by the formula F = ∇ × (4 yz 2 i + 3 x j + xz k ) . Calculate the ±ux of F upwards through the hemispherical surface S 1 . Copyright c c 2012 The University of Sydney...
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.

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