Chapter 14: Q. 35 (page 1132)

Verify Green’s 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’s Theorem) $\int_{C} F (x, y) . d r$ , 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,

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

Step by step solution

Step 1. Given Information.

We have to find the integral $\int_{C} F (x, y) . d r$ 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 $0 \leq t \leq 2 π$ .

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,

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Statistics

Calculus

Mechanics Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Verify Green’s 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’s Theorem) ∫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

Statistics

Calculus

Mechanics Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.