In order to apply Green’s theorem, the line integral of the boundary should be evaluated such...

In order to apply Green’s theorem, the line integral of the boundary should be evaluated such that the integration region inside the boundary lies always on the left as one advances in the direction of integration. What happens if the region lies on the right? How can you apply the theorem then? Explain.

Solutions

Expert Solution

Don't forget to hit tumb up if you like my explanation.

Dear student,

yes you are correct ..!! We have have the consider the integration region on left.

If you consider right you will end up getting an negative sign to your actual anwer.

For clarification ,

NOTE THAT THE REGION LIYING ON RIGHT AND LEFT HAS NO MEANING AT ALL.

in one situation if you get your domain on left while walking across the boundary,then to get domain on right you just reverse your direction.This simply means both cases are possible in any problem.but in order to use greens therom one must consider your direction of motion (which is nothing but orientation of the boundary) such that you get domain on left side.

Only we were said that,you walk around the surface such that the domain is on the left side.Then the direction up your head is normal direction.

If you are given with a case were right side domain is considerd then put -ve sign in greens therom and remaining all are same.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2...

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2 - 2x + y^2 = 0 , with the counterclockwise orientation. Don’t compute the FINAL TRIG integral. F(x,y) = < (-y^3 / 3) - cos(x^7) , cos(y^9 + y^5) + (x^3 / 3) > .

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2-2x+...

Using Green’s theorem, compute the line integral of the vector field below, along the curve x^2-2x+ y^2=0 , with the counterclockwise orientation. Don’t compute the FINAL TRIG integral. F(x,y)=<- y^3/3-cos⁡(x^7 ) ,cos(y^9+y^5 )+ x^3/3> .

Write down Green’s Circulation Theorem. Explain when Green’s Circulation Theorem applies and when it does not....

Write down Green’s Circulation Theorem. Explain when Green’s Circulation Theorem applies and when it does not. Give an example of Green’s Circulation Theorem showing the function, the integral and drawing the region.

Derive Green’s Theorem from Sturm-Liouville Eigen Value problem. When do you need Green’s theorem in the...

Derive Green’s Theorem from Sturm-Liouville Eigen Value problem. When do you need Green’s theorem in the context of a Heat or Diffusion Equation?

What is the relation between “Green’s Theorem” and “Stokes’s Theorem”? Explain about the transformations defined in...

What is the relation between “Green’s Theorem” and “Stokes’s Theorem”? Explain about the transformations defined in these theorems. What is the most important application and consequence of “Stokes’s Theorem”? Explain Independence of path?

Vector Analysis: Verify Green’s Theorem in the plane for ? ⃑ = (?^2 + ?^2)?̂+ (?^2...

Vector Analysis: Verify Green’s Theorem in the plane for ? ⃑ = (?^2 + ?^2)?̂+ (?^2 − ?^2)?̂ in the anti-clockwise direction around the ellipse 4?^2 + ?^2 = 16.

Problem 1: Use Green's Theorem to evaluate the vector line integral ∫C [?3??−?3] ?? where ?...

Problem 1: Use Green's Theorem to evaluate the vector line integral ∫C [?3??−?3] ?? where ? is the circle ?2+?2=1 with counterclockwise orientation. Problem 2: Which of the following equations represents a plane which is parallel to the plane 36?−18?+12?=30 and which passes through the point (3,6,1) ? a). 6?−3?+2?=3 b). 6?+3?−2?=34 c). 36?+18?−12?=204 d). 6?−3?+2?=2 e). 36?+18?+12?=228

Solve the following equation using Green’s functions. Set up the integral for the particular solution and...

Solve the following equation using Green’s functions. Set up the integral for the particular solution and evaluate y''+9y=x+sinx y(0)=1,y'(0)=2

Fundamental Theorem of Line Integrals Consider the line integral: ∮_C▒〈6xy+6x, 3x^2-3 cos⁡(y) 〉 ∙dr ⃗ where...

Fundamental Theorem of Line Integrals Consider the line integral: ∮_C▒〈6xy+6x, 3x^2-3 cos⁡(y) 〉 ∙dr ⃗ where C is the line segment from the origin to (4, 5). Evaluate this integral by rewriting the integral as a standard single variable integral of the parameter t. Without finding a potential function show that the integrand is a gradient field. Find a potential function for the vector field. Use the Fundamental Theorem of Line Integrals to evaluate this integral.

1) what are Cauchy’s Integral Theorem and Cauchy's integral formula 2) Explain about the consequences and...

1) what are Cauchy’s Integral Theorem and Cauchy's integral formula 2) Explain about the consequences and applications of these theorems. 3) Explain about different types of singularities and the difference between Taylor series and Laurent series.

Subjects