Set 13 Solutions.pdf - ESE 318-02 Fall 2018 Homework...

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ESE 318-02, Fall 2018 Homework Set #13 (14 problems) Due Tuesday, Dec. 4 Don’t use Wolfram -Alpha or computer to do your integrals. You may use a Table of integrals, but you shouldn’t need any other than the inside front cover of your text, and the hints below. 1 . Evaluate C dy y x dx xy x 2 2 2 , where C is the boundary of the region in the first quadrant bounded by 3 2 and x y x y . 2 . Zill 9.12.12. Assume counter-clockwise. Hint: integrate over y first. 3 . Zill 9.12.13. 4 . Zill 9.12.30. 5 . Evaluate , where C is the counter-clockwise boundary of the region in the first quadrant bounded by 2 2 2 and x y x y . 6 . Find the flux of F over the given surface, S . 8 8 1 0 3 4 , , : 0 , 5 , 2 v u v u v u S y x r F 7 . Find the flux of F over the given surface, S . v u u v u v u S z y x 4 0 , sin , cos : , , 2 r F 8 . Find the outward flux (away from origin) of xyz yze xz yz , , F through the portion of the surface S : y = cos x , where 1 0 and 0 2 z x . 9 . Zill 9.13.38. Do not use the Divergence Theorem. (Hint: note that the surfaces are flat. This is more of a thinking problem than a calculating one. Don’t just start evaluating integrals.) 10 . Find the rightward (positive y ) flux of the vector field F = xyz 2 i + y 2 j + y 3 e xz k , through the vertical disc: 3 , 4 7 5 2 2 y z x .
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ESE 318-02, Fall 2018 11 . In class I developed these two expressions. Differential Surface Area: dudv dS v u r r Flux Integral ( F a vector field):      R v u R S dudv v u dudv dS r r F N F n F , I also showed that for the case where the surface can be parameterized as z = f ( x , y ), dS and the flux integral can be expressed as follows: Differential Surface Area: dxdy f f dS y x 2 2 1 Flux Integral:   R S dxdy R y z Q x z P dS n F (See also Zill problem 9.13.43.) (a) Derive similar formulas for a surface parameterized as y = g ( x , z ). (Orient N so that it is positive in the y direction.) (b) Derive similar formulas for a surface parameterized as x = h ( y , z ). (Orient N so that it is positive in the x direction.) In the remaining problems, the surface can be parameterized as z = f ( x , y ). 12 . Zill 9.13.30. 13 . Zill 9.13.32. 14 . Zill 9.13.34. Hint: C e u du ue u u ) 1 ( . General tip on surface integrals: Take your time understanding the surface and region of integration. Draw the region of integration, which is always in the uv or xy plane for these problems. (Some 3D pictures may be difficult to draw/picture; what’s important is the region in the uv or xy plane.)
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ESE 318-02, Fall 2018 Solutions 1 . Evaluate C dy y x dx xy x 2 2 2 , where C is the boundary of the region in the first quadrant bounded by 3 2 and x y x y .
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