Q. 7 Explain how your answer to Exerc... [FREE SOLUTION]

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Chapter 14: Q. 7 (page 1131)

Explain how your answer to Exercise 6 is relevant to the discussion of Green鈥檚 Theorem.

Short Answer

Expert verified

It has been explained how the answer to Exercise 6 is relevant to the discussion of Green鈥檚 Theorem.

Step by step solution

Step 1. Given information.

The objective is to explain how the answer to Exercise 6 is relevant to the discussion of Green鈥檚 Theorem.

Step 2. Green's Theorem

Green's Theorem states that,
"Let be a region in the plane with a smooth boundary curve C oriented counterclockwise by

$r (t) = 鉄� (x (t), y (t)) 鉄� for a 鈮� t 鈮� b$

If a vector field $F (x, y) = <F_{1} (x, y), F_{2} (x, y)>$ is defined on R, then

${鈭�}_{C} 鈥� F 鈰� d r = {鈭�}_{R} 鈥� (\frac{鈭� F_{2}}{鈭� x} 鈭� \frac{鈭� F_{1}}{鈭� y}) d A 鈰� "$

Step 3. Explanation

Notice that, if you set, $G (x, y) = \frac{鈭� F_{2}}{鈭� x} 鈭� \frac{鈭� F_{1}}{鈭� y}$ , then you can use Green's Theorem to evaluate the integral ${鈭�}_{R} 鈥� G (x, y) d A$ .

Hence, on setting $G (x, y) = \frac{鈭� F_{2}}{鈭� x} 鈭� \frac{鈭� F_{1}}{鈭� y}$ , allows the use of Green鈥檚 Theorem to evaluate the integral.

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91影视

Explain how your answer to Exercise 6 is relevant to the discussion of Green鈥檚 Theorem.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Green's Theorem

Step 3. Explanation

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