vcprac06 - You should use whichever method you think will...

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The University of Sydney School of Mathematics and Statistics Practice Session 6 MATH2061: Vector Calculus Summer School 2012 1. Evaluate ii S F · n dS , where F = 2 x i +3 xy j yz 2 k and S is the surface bounded by the planes x = 0 , x + y = 2 , y = 0 , z = 0 and z = 2 . ( n is the unit outward normal to the surface S. ) 2. Find the ±ux of F = xz 2 i +( x 2 y z 3 ) j +(2 xy + y 2 z ) k outwards across the entire surface of the hemispherical region bounded by z = ( a 2 x 2 y 2 ) 1 / 2 and z = 0 . 3. Evaluate ii S ( ∇ × F ) · n dS , where S is the surface of the hemisphere x 2 + y 2 + z 2 = a 2 for z 0, F = y i + zx j + y k and n is the unit normal with positive k -component. (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.
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Unformatted text preview: You should use whichever method you think will be the easiest.) 4. Evaluate c C x 2 y 3 dx + dy + z dz , where C is x 2 + y 2 = a 2 , z = 0, taken once, in an anti-clockwise direction when viewed from above. 5. By using a suitable integration theorem, or otherwise, evaluate the line integral c C y dx + z dy + x dz where C is the intersection of the sphere x 2 + y 2 + z 2 = a 2 and the plane x + y + z = 0. Assume that C is oriented anticlockwise, when viewed from a point on the z-axis with z > 0. Copyright c c 2012 The University of Sydney...
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