The pair of options were stated to be incorrect, did I fail to include another option or simply chose the wrong pair? **Question:**Select all of the following curves $ C $ and vector fields $ \vec{F} $ for which Green's Theorem could be used to calculate $ \oint_C \vec{F} \cdot d\vec{r} $. Please note that multiple answers may be correct.1. $\vec{F}(x, y) = x^2 \vec{i} + y^2 \vec{j}$, and $ C $ is a circle of radius 1 in the xy-plane centered at $(0, 0)$ and oriented counterclockwise.2. $\vec{F}(x, y) = \vec{i} + 2 \vec{j} + 3 \vec{k}$, and $ C $ is the circle of radius 3 in the plane $ y = 4 $, centered at the point $(0, 4, 0)$ and oriented counterclockwise as viewed from the positive y-axis.3. $\vec{F}(x, y) = y^2 \vec{i} + x^2 \vec{j}$, and $ C $ is the boundary of the rectangle having vertices $(0, 0), (3, 0), (3, 2), (0, 2)$, oriented counterclockwise. **(Selected)**4. $\vec{F}(x, y) = x^2 \vec{i} + y^2 \vec{j}$, and $ C $ is the line segment from $(0, 0)$ to $(2, 3)$.5. $\vec{F}(x, y) = y^2 \vec{i} + x^2 \vec{j}$, and $ C $ is a circle of radius 1 in the xy-plane centered at $(0, 0)$ and oriented counterclockwise. **(Selected)**

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Description: The pair of options were stated to be incorrect, did I fail to include another option or simply chose the wrong pair? **Question:**Select all of the following curves $ C $ and vector fields $ \vec{F} $ for which Green's Theorem could be used to c

Answered: Select all of the following curves Cand…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The pair of options were stated to be incorrect, did I fail to include another option or simply chose the wrong pair?

Expert Solution

Step 1

The chosen pairs were correct but you failed to include the first option.

The correct answers are

(1) $\vec{F} (x, y) = x^{2} \vec{i} + y^{2} \vec{j}$ ,and $C$ is the circle of radius 1 in the $x y$ -plane centered at $(0, 0)$ and oriented counterclockwise.

(3) $\vec{F} (x, y) = y^{2} \vec{i} + x^{2} \vec{j}$ ,and $C$ is the boundary of the rectangle having vertices $(0, 0), (3, 0), (3, 2) a n d (0, 2),$ oriented counterclockwise.

(5) $\vec{F} (x, y) = y^{2} \vec{i} + x^{2} \vec{j}$ , and $C$ is the circle of radius 1 in the $x y$ -plane centered at $(0, 0)$ and oriented counterclockwise.

Green's theorem states that ,"Let $C$ be a positively oriented simple closed curve with interior region $R$ and assume that $C$ is piecewise smooth. If the vector field $\vec{F} = (M, N)$ is defined and differentiable on $R$ then

$\int_{C} M d x + N d y = \iint_{R} N_{x} - M_{y} d A$

In two dimensions , $c u r l \vec{F} = N_{x} - M_{y}$

So in vector form, Green's theorem is written as

$\int_{C} \vec{F} . d r = \iint_{R} c u r l \vec{F} d A$

Step 2

Green's theorem can be used only for the vector fields in two dimensions. It cannot be used for vector fields in three dimensions.

Therefore, option (2) is wrong.

Green's theorem can be used only if the curve $C$ is a simple closed curve. It cannot be used for line segment.

Therefore, option (4) is wrong.

In vector form, Green's theorem is written as

$\int_{C} \vec{F} . d r = \iint_{R} c u r l \vec{F} d A$ , where $c u r l \vec{F} = N_{x} - M_{y}$

First for option (1) , $\vec{F} (x, y) = x^{2} \vec{i} + y^{2} \vec{j}$ ,and $C$ is the circle of radius 1 in the $x y$ -plane centered at $(0, 0)$ and oriented counterclockwise.

Here,

$M = x^{2} a n d N = y^{2} M_{y} = 0 a n d N_{x} = 0$

$c u r l \vec{F} = N_{x} - M_{y} = 0$

Here, $x v a r i e s f r o m 0 \to 1$

and $y v a r i e s f r o m 0 \to 1$

Therefore, $\int_{C} \vec{F} . d r = \int_{0}^{1} \int_{0}^{1} (0) d x d y = 0$

Therefore option (1) is correct.

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