265Lesson25Problems(1)

265Lesson25Problems(1) - - G ( x,y,z ) = h-y,-z,-x i ? (5)...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 25 PROBLEMS Problems are worth 20 points each. (1) Compute the divergence of the vector field -→ F ( x,y,z ) = h x + 3 z, 3 x - 2 y, 2 x + z i . (2) Sketch the solid described by x 2 + y 2 z 1. Use the Divergence Theorem to evaluate the surface integral over the boundary of that solid of the vector field -→ F ( x,y,z ) = y -→ ı + z -→ + xz -→ k . (3) Use the Divergence Theorem to compute the surface integral ZZ T -→ F · d -→ S where T is the unit sphere x 2 + y 2 + z 2 = 1 and -→ F ( x,y,z ) = h y,z,x i . (4) What would be the value of the surface integral in problem (3) if instead of -→ F , we used the vector field
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Unformatted text preview: - G ( x,y,z ) = h-y,-z,-x i ? (5) Use the Divergence Theorem to calculate the ux of- F ( x,y,z ) = h x 3 ,y 3 ,-3 z 2 i through the boundary of the solid T given by x 2 + y 2 4 , z 2. Assuming- F is the velocity eld of a uid, is the net ow into the solid T , or out of T ? (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Show that the divergence of the curl of a vector eld (assuming all derivative mentioned exist) is always 0. 1...
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