hmwk10 - y = x 2 z 2 and in the half-space x y z | y ≤ 1...

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Math 215 Homework Set 10: §§ 17.7 – 17.8 Winter 2008 Most of the following problems are modified versions of the recommended homework problems from your text book Multivariable Calculus by James Stewart. 17.7a A fluid with density 45 grams per cubic centimeter flows with velocity v ( x, y, z ) = y, 1 , z . Find the rate of flow outward through the paraboloid z = 16 - ( x 2 + y 2 ) / 9 for x 2 + y 2 144 . 17.7b Find the flux of F ( x, y, z ) = y, 2 x, 3 z across the surface of the cube with vertices ( ± 1 , ± 1 , ± 1) . 17.7c Find the center of mass of the hemisphere given by the equations x 2 + y 2 + z 2 25; y 0 . Assume the density of the hemisphere is constant. 17.7d Find the flux of F ( x, y, z ) = xy, y 2 , z across the surface S where S is the part of the paraboloid z = 2 x 2 + 2 y 2 below the plane z = 2 with downward orientation. 17.8a. Verify that Stokes’ Theorem is true for the vector field F ( x, y, z ) = x, z 2 , y 2 for the surface that lies on the paraboloid
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Unformatted text preview: y = x 2 + z 2 and in the half-space { ( x, y, z ) | y ≤ 1 } . We assume that the surface is oriented in the positive direction of the y-axis. 17.8b. Suppose S is a surface with boundary C . Show that if v is a fixed vector and F ( x, y, z ) = ± x, y, z ² , then 2 ± ± S v · d S = ± C ( v × F ) · d r . 17.8c. Suppose F is a vector field. Consider two surfaces S and S ± which possess the same boundary C . Use sketches (note plural) to describe how S and S ± must be oriented so that ± ± S curl( F ) · d S = ± ± S ± curl( F ) · d S ± . 17.8d. Suppose F is a vector field. Expanding upon the above problem: Suppose that S is a closed surface (that is, S is a boundary for a region of space). Show that ± ± S curl( F ) · d S = 0 ....
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