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Chapter 6: Q22MP (page 338)

鈭�V脳苍诲蟽over the entire surface of the sphere, iflocalid="1657353129148" (x鈭�2)2+(y+3)2+(z)2=9, ifV=(3x鈭�yz)i+(z2鈭�y2)j+(2yz+x2)k

Short Answer

Expert verified

The solution is鈭�鈭�T鈥�V脳苍诲蟽=108蟺.

Step by step solution

01

Given Information.

V=(3x鈭�yz)i+z2鈭�y2j+2yz+x2k

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 is the surface area that encloses the volume T.

鈭�脳V=鈭�Vx鈭�x+鈭�Vy鈭�y+鈭�Vz鈭�z

=3-2y+2y

=3

Here,鈭�蟽is the entire surface of the sphere of radius 2centered at (2,-3,0), soT is just the volume of the sphere.

鈭�鈭�T鈥�V脳苍诲蟽=鈭�鈭�T鈥�鈥�鈭�脳VdT

=3鈭�T鈥�dT

=343(蟺)(3)3

=108蟺

Hence, the solution is .鈭�鈭�T鈥�V脳苍诲蟽=108蟺

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Most popular questions from this chapter

Find vector fields Asuch that V=curlAfor each givenV=i(zezy+xsin鈦�zx)+jxcos鈦�xz鈭�kzsin鈦�zx

Evaluate each of the following integrals in the easiest way you can.

鈭�(2ydx-3xdy)around the square bounded by x=3,x=5,y=1andy=3.

If the temperature is T=x2-xy+z2, find

(a) The direction of heat flow at (2,1, -1);

(b) The rate of change of temperature in the direction j-kat(2,1,-1)

鈭�V脳苍诲蟽over the closed surface of the tin can bounded by if x2+y2=9,z=0,z=5ifV=2xyi鈭�y2j+(z+xy)k.

鈭�鈥�curlV)脳nd蟽over any surface whose bounding curve is in the (x,y)plane, where V=(x鈭�x2z)i+(yz3鈭�y2)j+(x2y鈭�xz)k.
See all solutions

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