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Chapter 6: Q6P (page 314)

For a simple closed curve Cin the plane show by Green鈥檚 theorem that the area inclosed is

A=12鈭�C(xdy-ydx)

Short Answer

Expert verified

The solution to this problem is that the condition is satisfied and the area is inclosed.

Step by step solution

01

Given Information.

The given information is A=12鈭�C(xdy-ydx).

02

Definition of Green’s Theorem

The 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.

Area of a simple curve is obtained by using the formula below.

鈭�dydx

Compare the area formula鈥檚 equation to Green鈥檚 Theorem.

role="math" localid="1659153988451" 鈭�Q鈭�z-鈭�P鈭�y=1

Find the differentiation with respect to.

P=-yxQ=x2

Use the value of P and Q and find the value of their derivatives with respect to z and y, respectively.

鈭�Q鈭�z=12鈭�P鈭�y=-12

The values of the derivatives satisfy the condition.

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Most popular questions from this chapter

Evaluate鈭�V鈰�nd蟽over the curved surface of the hemispherex2+y2+z2=9,z鈮�0, ifV=yi+xzj+(2z-1)k.Careful: See Problem 9.

Show by the Lagrange multiplier method that the maximum value of 诲蠄/dsis|诲蠄|.That is, maximize 诲蠄/dsgiven by (6.3) subject to the condition a2+b2+c2=1. You should get two values (卤) for the Lagrange multiplier 位, and two values (maximum and minimum) for诲蠄/dswhich is the maximum and which is the minimum?

Find the derivative of zexcosy at (1,0,蟺3) in the direction of the vector i+2j .

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.

Write out the equations corresponding to (9.3) and (9.4) for Q=2Xbetween points 3 and 4 in Figure 9.2, and add them to get (9.6).
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