In this lesson, we will learn how to apply Green’s theorem to evaluate a line integral around a closed curve as the double integral over the plane region bounded by the curve.

Q1:

Use Greenβs theorem to determine the conditions on π, π, π, and π for the vector field F(π₯,π¦)=β¨ππ₯+ππ¦,ππ₯+ππ¦β© to be conservative. In that case, what is the potential function π(π₯,π¦) for F that satisfies π(0,0)=0?

Q2:

Use Greenβs theorem to find οΈβ ο’Frd, where πΆ is the circle of radius π and center at the origin and F(π₯,π¦)=β¨2π₯+5π¦,2π₯+7π¦β©.

Q3:

The figure shows the graph of π(π₯)=β3οΌπ₯+13ο(π₯β1) over the interval [0,1]. Let π be the shaded region and πΆ its boundary, traced counterclockwise. Let Fij(π₯,π¦)=π¦+π¦.

Use Greenβs theorem to calculate ο β ο’Frd.

Calculate οΈβ ο’ο Frd, where πΆο§ is the line from π to π.

Calculate οΈβ ο’ο‘Frd, where πΆο¨ is the curve from π to π.

Calculate οΈβ ο’ο’Frd, where πΆο© is the line from π to π.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.