265Lesson25Problems(1)

# 265Lesson25Problems(1) - -→ G x,y,z = h-y-z-x i(5 Use the...

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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 ﬂux 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 ﬁeld of a ﬂuid, is the net ﬂow 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 ﬁeld (assuming all derivative mentioned exist) is always 0. 1...
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## This note was uploaded on 02/24/2011 for the course MATH 265 taught by Professor Jerrymetzger during the Winter '11 term at North Dakota.

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