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Chapter 6: Q2P (page 313)

鈭�2xdy-3ydxaround the square with vertices(0,2),(2,0),(-2,0),(0,-2).

Short Answer

Expert verified

The integral will give the solution to be 40.

Step by step solution

01

Given information

The vertices of the square are (0,2),(2,0),(-2,0,(0,-2)

02

Definition of conservative force and scalar potential

A force is said to be conservative if 鈭�脳F=0..

The scalar potential is independent of the path. The scalar potential is the sum of potential in all the 3 dimensions calculated separately.

The formula for the scalar potential is W=鈭�F.dr..

03

Use substitution.

The green theorem in a plane is mentioned.

鈭�鈭�Q鈭�x-鈭�P鈭�y)=鈭�Pdx+Qdy

Takethe values as mentioned below.

P=-3yQ=2x

Substitute the value.

鈭�Q鈭�x-鈭�P鈭�y=2+3鈭�Q鈭�x-鈭�P鈭�y=5

04

Use Greens Theorem.

Use the Green theorem.

-3xdx+2ydy5鈭�dxdy5鈭�08鈭�08dxdy5鈭�088dx58840

Hence the integral 鈭�2xdy-3ydx gives 40

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Most popular questions from this chapter

Given that B=curlA, use the divergence theorem to show that over any closed surface is zero.

Given the vector.A=(x2鈭�y2)i+2xyj

(a) Find 鈭�脳A.

(b) Evaluate鈭�(鈭�脳A)脳诲蟽 over a rectangle in the(x,y) plane bounded by the lines x=0,x=a,y=0,y=b.

(c) Evaluate around the boundary of the rectangle and thus verify Stokes' theorem for this case.

Use either Stokes' theorem or the divergence theorem to evaluate each of the following integrals in the easiest possible way.

Givenu=xy+yz+zsinx, find

(a) 鈭�uat(0,1,2);

(b) The directional derivative of (0,1,2) at in the direction 2i+2j-k;

(c) The equations of the tangent plane and of the normal line to the level surface u=2at(0,1,2);

(d) a unit vector in the direction of most rapid increase of u at(0,1,2)

Evaluate each of the integrals in Problems3to 8as either a volume integral or a surface integral, whichever is easier.

鈭�鈭�脳痴诲蟿over the volumerole="math" localid="1657334446941" x2+y2鈮�4,0鈮�z鈮�5,V=(x2+y2)(ix+jy)

Verify that the force field is conservative. Then find a scalar potential 胃 such that F=-鈭�蠁,F=z2sinhyj+2zcoshyk,k=constant.
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