Homework7

Homework7 - given by v = i + x j + z k in meters/second....

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1 Mathematics 1c: Homework Set 7 Due: Monday, May 24 at 10am. 1. (10 Points) Section 7.3, Exercise 6 . Find an expression for a unit vector normal to the surface x = 3 cos θ sin φ, y = 2 sin θ sin φ, z = cos φ for θ in [0 , 2 π ] and φ in [0 ] . 2. (15 Points) Section 7.3, Exercise 15 (a) Find a parameterization for the hyperboloid x 2 + y 2 - z 2 = 25 . (b) Find an expression for a unit normal to this surface. (c) Find an equation for the plane tangent to the surface at ( x 0 ,y 0 , 0) , where x 2 0 + y 2 0 = 25 . (d) Show that the pair of lines ( x 0 ,y 0 , 0) + t ( - y 0 ,x 0 , 5) and ( x 0 ,y 0 , 0) + t ( y 0 , - x 0 , 5) lie in the surface and as well as in the tangent plane found in part (c) . 3. (10 Points) Section 7.4, Exercise 6 . Find the area of the portion of the unit sphere that is cut out by the cone z p x 2 + y 2 . 4. (10 Points) Section 7.5, Exercise 2 . Evaluate ZZ S xyz dS where S is the triangle with vertices (1 , 0 , 0) , (0 , 2 , 0) and (0 , 1 , 1) . 5. (10 Points) Section 7.6, Exercise 7 . Calculate the integral ZZ S F · d S , where S is the surface of the half-ball x 2 + y 2 + z 2 1 ,z 0 , and where F = ( x + 3 y 5 ) i + ( y + 10 xz ) j + ( z - xy ) k . 6. (10 Points) Section 7.6, Exercise 15. Let the velocity field of a fluid be
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Unformatted text preview: given by v = i + x j + z k in meters/second. How many cubic meters of uid per second are crossing the surface x 2 + y 2 + z 2 = 1 ,z 0? (Distances are in meters.) 7. (15 Points) Section 7.6, Exercise 18 . If S is the upper hemisphere { ( x,y,z ) | x 2 + y 2 + z 2 = 1 ,z } oriented by the normal pointing out of the sphere, compute ZZ S F d S for parts (a) and (b) . 2 (a) F ( x,y,z ) = x i + y j (b) F ( x,y,z ) = y i + x j (c) for each of the vector elds above, compute ZZ S ( F ) d S and Z C F d S , where C is the unit circle in the xy plane traversed in the counterclock-wise direction (as viewed from the positive z axis). (Notice that C is the boundary of S . The phenomenon illustrated here will be studied more thoroughly in the next chapter, using Stokes theorem.)...
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

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Homework7 - given by v = i + x j + z k in meters/second....

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