Chapter 6: Q6P (page 314)

For a simple closed curve Cin the plane show by Green’s theorem that the area inclosed is
$A = \frac{1}{2} \oint_{C} (x d y - y d x)$

Short Answer

Expert verified

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

Step by step solution

Given Information.

The given information is $A = \frac{1}{2} \oint_{C} (x d y - y d x)$ .

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.

Find the solution.

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

$\iint d y d x$

Compare the area formula’s equation to Green’s Theorem.

role="math" localid="1659153988451" $\frac{\partial Q}{\partial z} - \frac{\partial P}{\partial y} = 1$

Find the differentiation with respect to.

$P = \frac{- y}{x} Q = \frac{x}{2}$

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

$\frac{\partial Q}{\partial z} = \frac{1}{2} \frac{\partial P}{\partial y} = \frac{- 1}{2}$

The values of the derivatives satisfy the condition.

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For a simple closed curve Cin the plane show by Green’s theorem that the area inclosed is
$A = \frac{1}{2} \oint_{C} (x d y - y d x)$

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem

Find the solution.

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For a simple closed curve Cin the plane show by Green’s theorem that the area inclosed isA=12∮C(xdy-ydx)

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Famous Physicists

Physical Quantities And Units

Space Physics

Mechanics and Materials

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

For a simple closed curve Cin the plane show by Green’s theorem that the area inclosed is
$A = \frac{1}{2} \oint_{C} (x d y - y d x)$