# What is Green's theorem: Definition and 134 Discussions

In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

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27. ### A question about path orientation in Green's Theorem

So if we have a non-simply-connected region, like this one to apply Green's Theorem we must orient the C curves so that the region D is always on the left of the curve as the curve is traversed. Why is this? I have seen some proofs of Green's Theorem for simply connected regions, and I...
28. ### Green's Theorem vs Fundamental

Homework Statement 1) How do I know when to use Green's Theorem, the Fundamental Theorem for Line Integrals or the regular method of using parametrization? 2) Assuming that the three methods above are all used to solve line integrals, why do the Fundamental Theorem and Green give different...
29. ### Using Green's Theorem for line integral

Homework Statement Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 1) Is the statement above the same as finding the area enclosed? 2) ##\int_C \cos ydx + x^2\sin ydy ##, C is the rectangle with vertices (0,0) (5,0) (5,2) and (0,2). 3) ##\int_C y^4...
30. ### Green's Theorem to find Area help

Homework Statement Find the area of the right leaf of the Lemniscate of Gerono (the ∞ sign, see figure below) parametrized by r(t)= <sin(t), sin(t)cos(t)> from 0=<t=<pi Picture is uploaded. Homework Equations Green's theorem: integral of fdx+gdy = double integral (over the region) of (gx-fy)...
31. ### MHB Solving an IVP with Green's Theorem: Wondering?

Hey! :o If we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x\geq 0 \\ u_t(x,0)=g(x)$$ and we want to apply Green's theorem do we have to expand the problem to $x \in \mathbb{R}$ ?? (Wondering)
32. ### Using Green's Theorem for Vector Fields

Homework Statement Homework Equations n/a The Attempt at a Solution I don't understand how the book went from calculating Green's theorem on ##\int _c Pdx + Qdy + \int _{-c'} Pdx + Qdy = ## (1 in the attached picture) to getting (labeled 2) ##\int _c Pdx + Qdy = \int _{c'} Pdx + Qdy ##...
33. ### MHB Differential equation - Green's Theorem

Hey! :o I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ using Green's theorem but I got stuck... I found the following example in my notes...
34. ### Using the Divergence Theorem to Prove Green's Theorem

Homework Statement Prove Green's theorem \int_{\tau} (\varphi \nabla^{2} \psi -\psi\nabla^{2}\varphi)d\tau = \int_{\sigma}(\varphi\nabla\psi -\psi\nabla\varphi)\cdot d\vec{\sigma} Homework Equations div (\vec{V})=\lim_{\Delta\tau\rightarrow 0} \frac{1}{\Delta\tau} \int_{\sigma} \vec{V} \cdot...
35. ### Green's Theorem and simple closed curve

Homework Statement Use, using the result that for a simple closed curve C in the plane the area enclosed is: A = (1/2)∫(x dy - y dx) to find the area inside the curve x^(2/3) + y^(2/3) = 4 Homework Equations Green's Theorem: ∫P dx + Q dy = ∫∫ dQ/dx - dP/dy The Attempt at a Solution I...
36. ### Green's Theorem homework problem

Homework Statement Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (-2,1), (-2,-3), (1,-1), (1,5) and back to (-2,1), in that order. Use Green's Theorem to evaluate the integral ∫(2xy)dx+(xy2)dy. Homework Equations Green's Theorem: ∫Pdx+Qdy...
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### Understanding Green's Theorem in 2-Dimensional Vector Fields

Hi PF! So let's say I have some vector field, call it ##\vec{F}## and let ##\vec{F}## be 2-Dimensional and suppose I wanted to compute ##\iint_D \nabla \cdot \vec{F} dD##. Using green's theorem we could write ##\iint_D \nabla \cdot \vec{F} dD = - \int_{\partial D} \vec{F} \cdot \hat n dS##...

49. ### Green's theorem, relation between two integrals

Homework Statement . Calculate by a line integral the following double integral: ##\iint\limits_D (y^{2}e^{xy}-x^{2}e^{xy})dxdy##, D being the unit disk. The attempt at a solution. Well, if we consider C to be the curve that encloses the region D (C is the unit circle), then C is a...
50. ### Direct Integration vs. Green's Theorem

Problem: ##\int_R (x-y)dx \ dy=-2/3 ## for ##R=\{(x,y):x^2+y^2 \geq 1; y \geq 0\}## by a.) Direction integration, b.) Green's theorem. Attempt at a Solution: I'm a little confused with part a. Wouldn't the region R be defined by all the points above the y-axis that lie on, in addition to...