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Chapter 6: Q23MP (page 338)

鈭�F脳苍诲蟽where F=(y2鈭�x2)i+(2xy鈭�y)j+3zkand 蟽is the entire surface of the tin can bounded by the cylinder

role="math" localid="1657353627256" x2+y2=16

role="math" localid="1657353639412" z=3

role="math" localid="1657353647648" z=-3

Short Answer

Expert verified

The solution is 鈭�鈭�T鈥�F脳苍诲蟽=192蟺.

Step by step solution

01

Given Information.

F=y2鈭�x2i+(2xy鈭�y)j+3zk

02

Definition of Divergence Theorem.

The divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed.

03

Find the solution.

Use the divergence theorem 鈭�T鈥�鈭�脳VdT=鈭�鈭�T鈥�V脳苍诲蟽, where鈭�Tis the surface area that encloses the volume T.

鈭�脳F=鈭�Fx鈭�x+鈭�Fy鈭�y+鈭�Fz鈭�z

=鈭�2x+(2x鈭�1)+3

=2=2

Here, 鈭�蟽is the entire surface of the tin, and can be bounded by the cylinder.

x2+y2=16

z=3

z=-3

The tin鈥檚 radius and height are 4and 6respectively.

鈭�鈭�T鈥�F脳苍诲蟽=鈭�T鈥�鈭�脳FdT

=2鈭�T鈥�dT

=(2蟺)(4)2(6)

=192蟺

Hence, the solution is 鈭�鈭�T鈥�F脳苍诲蟽=192蟺.

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