Question

Question 1 Please

In a bathtub, the velocity of water near2 the drain is given by the vector field k: cm/sec (22 +1)2(22 +1)222 +1 where x, y, and z are measured in centimeters and (0, 0,0) is at the center of the drain 1. Rewriting F as follows, describe in words how the water is moving: (22 +1)2'(22 +1)2 2+1 Consider cach of the threc terms in cquation (4). (Look at some plots.) For fixed z, what is the flow like? How does the flow change with z'? 2. The drain in the bathtub is a disk in the ry-plane with center at the origin and radius 1 cm. Compute the flux of F through this disk, oricnted down. Give units for your answer. What is the meaning of this number (in tcrms of the tub watcr)? 3. Find the divcrgence of F. (Simplify your answer, if possiblc.) 4. Find the net flux of water out through the surface of a sphere of radius 1 centered at the origin, oriented outward. You may compute the flux integral directly or use arn appropriate theorem. 5. Set up and evaluate a flux integral to find the net flux of water through the bottom half of the sphere, -1-2y2, oriented downwards (Set up and evaluate the double integral in spherical coordinates; do not use a theorem here.) 1 - r2 - y2, oriented downwards. 6. For thc vector ficld G - G 5ん), compute where C is the edge of the drain oriented clockwise when viewed from above. 7. Calculate curlG. 8. Explain why your answers to questions 5 and 6 should be the same. (Cite a theorem.) 9. Explain why your answers to questions 2 and 5 should be the same. (Cite a theorem.) 2specifically, in the region where x2 +y2-1 and-1 < < 1

Answer 1

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