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Chapter 6: Q29MP (page 338)

鈭�V脳draround the boundary of the square with vertices(1,0),(0,1),(鈭�1,0),(0,鈭�1) , ifV=x2i+5xj

Short Answer

Expert verified

The solution is鈭�V脳dr=10

Step by step solution

01

Given Information.

The given information is,V=x2i+5xj

02

Definition of Green’s Theorem.

Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

03

Find the solution.

For, V=x2i+5xj

鈭�V脳dx=鈭�x2dx+5xdy

Use Green鈥檚 theorem 鈭�A鈥�鈭�Q鈭�x鈭�鈭�P鈭�ydxdy=鈭�鈭�APdx+Qdy, where 鈭�Ais the boundary of the area A.

It can be seen that

P=x2鈫�鈭�P鈭�y

=0

role="math" Q=5x鈫�鈭�Q鈭�x

=5

Since the contour is the boundary of the square with the given vertices, the area Ais then the area of the square.

鈭�V脳dr=5(1鈭�0)2+(0鈭�1)22

=10

Hence, the solution is 鈭�V脳dr=10.

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Most popular questions from this chapter

Evaluatethelineintegral鈭�(x2-y2)dx-2xydyalongeachofthefollowingpathsfrom(0,0)to(1,2).(a)y=2x2.(b)x=t2(c)y=0fromx=0tox=2;thenalongthestraightlinejoining(2,0)to(1,2).

Show thatF=y2zsinh鈦�(2xz)i+2ycosh2鈦�(xz)j+y2xsinh鈦�(2xz)k is conservative, and find a scalar potential 蠁such that F=鈭�鈭�蠒.

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.

鈭�V鈭�苍诲蟽over the curved part of the hemisphere in Problem 24, if role="math" localid="1657355269158" V=curl(yi鈭�xj).

Verify that the force field is conservative. Then find a scalar potential 蠁 such thatF=-鈭�蠁,F=yi+xj+k ,

K = constant.
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