Q. 35 Verify Green鈥檚 Theorem by work... [FREE SOLUTION]

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Chapter 14: Q. 35 (page 1132)

Verify Green鈥檚 Theorem by working in pairs. In each pair, evaluate the desired integrals first directly and then using the theorem:
Directly compute (i.e., without using Green鈥檚 Theorem) ${鈭�}_{C} F (x, y) . d r$ , where $F (x, y) = y^{2} i - x y j$ and $C$ is the unit circle traversed counterclockwise.

Short Answer

Expert verified

The required integral is,

${鈭�}_{C} F (x, y) . d r = 0$ .

Step by step solution

Step 1. Given Information.

We have to find the integral ${鈭�}_{C} F (x, y) . d r$ using Green's Theorem where $F (x, y) = y^{2} i - x y j$ .

Step 2. Finding the derivative.

The curve $C$ is the unit circle traversed counterclockwise, so this curve is parameterized as follows:

$r (t) = (\cos (t), \sin (t))$ for $0 鈮� t 鈮� 2 蟺$ .

Then, its derivative will be,

$r' (t) = \frac{d}{d t} (r (t)) = \frac{d}{d t} (\cos (t), \sin (t)) = (\frac{d}{d t} (\cos (t)), \frac{d}{d t} (\sin (t))) = (- \sin (t), \cos (t))$

Rewriting the vector field as follows:

$F (x, y) = y^{2} i - x y j F (x (t), y (t)) = \sin^{2} (t) i - \cos (t) \sin (t) j = (\sin^{2} (t), - \cos (t) \sin (t))$

Step 3. Evaluating the integral

Finding the required integral,

${鈭�}_{C} F (x, y) . d r = {鈭�}_{C} F (x (t), y (t)) . r' (t) d t = {鈭�}_{0}^{2 蟺} (\sin^{2} (t), - \cos (t) \sin (t)) . (- \sin (t), \cos (t)) d t = {鈭�}_{0}^{2 蟺} (- \sin^{3} (t) - \sin (t) \cos^{2} (t)) d t = {鈭�}_{0}^{2 蟺} - \sin (t) (\sin^{2} (t) + \cos^{2} (t)) d t = {鈭�}_{0}^{2 蟺} - \sin (t) d t = {[\cos (t)]}_{0}^{2 蟺} = \cos (2 蟺) - \cos (0) = 1 - 1 = 0$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the derivative.

Step 3. Evaluating the integral

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91影视

Verify Green鈥檚 Theorem by working in pairs. In each pair, evaluate the desired integrals first directly and then using the theorem:Directly compute (i.e., without using Green鈥檚 Theorem) 鈭�CFx,y.dr, where Fx,y=y2i-xyjand Cis the unit circle traversed counterclockwise.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the derivative.

Step 3. Evaluating the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Probability and Statistics

Pure Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.