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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  1. chwala

    Verify Green's Theorem in the given problem

    My lines are as follows; ##y=\sqrt x## and ##y=x^2## intersect at ##(0,0## and ##(1,1)##. Along ##y=\sqrt x##, from ##(0,0)## to ##(1,1)## the line integral equals, $$\int_0^1 [3x^2-8x] dx + \dfrac{4\sqrt x-6x\sqrt x}{2\sqrt x} dx $$ $$=\int_0^1[3x^2-8x+2-3x]dx=\int_0^1[3x^2-11x+2]dx =...
  2. WMDhamnekar

    Is the Calculation of the Vector Line Integral Over a Square Correct?

    Author's answer: Recognizing that this integral is simply a vector line integral of the vector field ##F=(x^2−y^2)i+(x^2+y^2)j## over the closed, simple curve c given by the edge of the unit square, one sees that ##(x^2−y^2)dx+(x^2+y^2)dy=F\cdot ds## is just a differentiable 1-form. The...
  3. A

    Green's theorem with a scalar function

    Greetings! My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!
  4. R

    I Double integral and Green's theorem

    Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region
  5. T

    Green's Theorem in 3 Dimensions for non-conservative field

    Homework Statement C is the directed curve forming the triangle (0, 0, 0) to (0, 1, 1) to (1, 1, 1) to (0, 0, 0). Let F=(x,xy,xz) Find ∫F·ds. Homework EquationsThe Attempt at a Solution My intial instinct was to check if it was conservative. Upon calculating: ∇xF=(0,-z,y) I concluded that...
  6. M

    Using Green's Theorem for a quadrilateral

    Homework Statement Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful. Homework Equations Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
  7. M

    Question about finding area using Green's Theorem

    Homework Statement Use Green's Theorem to find the area of the region between the x-axis and the curve parameterized by r(t)=<t-sin(t), 1-cos(t)>, 0 <= t <= 2pi Attached is a figure pertaining to the question Homework Equations [/B] The Attempt at a Solution Using the parameterized...
  8. M

    How to find the area of a triangular region using Green's Theorem

    Homework Statement You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree. Homework Equations Line integral of Pdx + Qdy = Double integral of...
  9. M

    MHB What is the correct way to apply Green's Theorem in this scenario?

    Hey! :o I want to compute the integral $\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy$, where $C$ is the boundary of the bounded field that is defined by the curves $y=2-x^2$ and $y=x$, with positive orientation. We have to apply Green's Theorem, or not...
  10. A

    Classical Please recommend two textbookss about the Poisson equation and Green's function

    Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
  11. U

    MHB Green's Theorem with Singularities

    So here's the question: You are given that F is a conservative vector field, except for singularities at the points (0,1), (2,0), (3,0), and (0,4). You are given the following information about line integrals around the following closed paths: 1) Around the curve C1 given by x^2 + y^2 = 2...
  12. davidge

    I Green's theorem in tensor (GR) notation

    Hi. I was trying to translate the divergence theorem and the Green's theorem to tensor notation that we use in Relativity. For the divergence theorem, it was easy (please tell me if I'm wrong in the below derivation). I'm using the standard electromagnetic tensor ##F_{\mu \nu}## in place of the...
  13. nysnacc

    Applying Green's Theorem: Solving Parametrized Homework Problems

    Homework Statement Homework Equations Green's theorem The Attempt at a Solution DO I first parametrize? For 1st part, I have 3 parametrizations, which I can then find the normal vector, and use in the integrals?
  14. i_hate_math

    Area of Region Vector Calculus

    I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...
  15. Alexstrasza

    How Can I Simplify This Integral Using Green's Theorem?

    Homework Statement I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy y≥0 2x=x^2+y^2 I used Green's theorem and got: ∬ (e^xcosy+2x) - (e^xcosy) dy dx x bounds: from 0 to 2 y bounds: from 0 to sqrt(2x-x^2) After solving all that stuff I get to: ∫ (2x) (sqrt(2x-x^2)) dx x bounds: 0 to...
  16. K

    I Green's theorem and Line Integrals

    (Sorry for my bad English.) I was reading about the Green's theorem and I notice that the book only shows for the case where the function is a vector function. When learning about line integrals, I saw that we can do line integrals using "ordinary" functions. For example, the line integral of...
  17. Y

    Potential and charge on a plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  18. Y

    Potential and total charge on plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  19. Rmehtany

    How Can One Solve This Complex Trigonometric Integral Analytically?

    Hey Guys! I was working on an integration problem, and I "simplified" the integral to the following: $$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(Nx)) \cdot (Z-v \cos(x)) dx$$ This integral may seem impossible (I have almost lost all hope on doing this...
  20. J

    Green's first identity at the boundary

    As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω. How about if the function is just discontinuous at the boundary? Actually, this function is an electric field. So its tangential component is naturally continuous, but the...
  21. Conservation

    Circulation of a triangular region

    Homework Statement Find the circulation (line integral) of y2dx+x2dy for the boundary of a triangular region contained within x+y=1, x=0, and y=0. Homework Equations Green's theorem The Attempt at a Solution I think I actually already got the solution; I used the Green's theorem to get the...
  22. J

    Green's first identity at the boundary

    As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω. How about if the function is just discontinuous at the boundary? For example, I intend to make a volume integration of a product of electric fields, the field function is...
  23. F

    Verify Green's Theorem in the plane for....

    Homework Statement Use Green''s Theorem in the plane to check: \oint_C (xy+y^2) \> dx + x^2 \> dy Where C is the closed curveof the region bound between the curve of y=x^2 and the line y=x Homework Equations \oint_C u \> dx + v \> dy = \int \int_A (\partial_x v - \partial_y u) \> dx \> dy...
  24. SquidgyGuff

    Calcularing area vector using line integral

    Homework Statement A closed curve C is described by the following equations in a Cartesian coordinate system: where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula: 2. The...
  25. kostoglotov

    Can Green's Theorem disagree with itself sometimes?

    Homework Statement Firstly, I was seeking any clarification on whether I've made any mistakes. Secondly, further insight into Green's Theorem, if my working is all good. I've been reading the mathinsight.org on Subtleties about curl: http://mathinsight.org/curl_subtleties Regarding the...
  26. M

    MHB Green's theorem - Boundary value problem has at most one solution

    Hey! :o Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\partial{u}}{\partial{y}}\right ) -(1+x^2+y^4)u=f(x,y), x^2+y^2<1 \\ u(x, y)=g(x,y)...
  27. kostoglotov

    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. Calpalned

    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. Calpalned

    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. RJLiberator

    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. M

    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. Calpalned

    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. M

    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. B3NR4Y

    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. Amy Marie

    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. L

    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...
  37. M

    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##...
  38. evinda

    MHB How do I calculate and use the different formulas of Green's Theorem?

    Hello! (Wave) I have a question.. There are three formulas of the Green Theorem: $$\oint_S (Mdx+Ndy)=\iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$ $$\oint \overrightarrow{F} \cdot d \overrightarrow{R}=\iint_R \nabla \times \overrightarrow{F}...
  39. G

    Green's Theorem Homework - Solve for ∫c (3x2 -1).dr

    Homework Statement Hello, I know this might be trivial but, can you please tell me what I am missing? Here is my problem : Homework Equations Given F = (yi+x3j).dr Q = x3 P = y => ∂Q/∂x = 3x2 and ∂P/dy = 1 ∫c (3x2 -1).dr = ∫2Pi0∫10 (3r2cosθ2-1) rdrdθ ∫2Pi0∫10 (3r3cosθ2-r) drdθ Is it true...
  40. Feodalherren

    Green's theorem - area of a cycloid

    Homework Statement Use Green’s Theorem to find the area of the region between the x – axis and one arch of the cycloid parameterized by p(t) = < t-2sin(t),2-2cos(t)> for 0≤t≤2∏ p Homework Equations The Attempt at a Solution My problem here is that I get different answers depending on if I...
  41. Feodalherren

    Green's theorem - confused about orientation

    Homework Statement ∫Fdr Over C where C is the cirlce (x-3)^2+ (y+4)^2=4 F=<y-cosy, xsiny> Homework Equations The Attempt at a Solution So I applied Green's theorem and converted to polar and ended up with -4π, it should be positive. The orientation confused me since day one...
  42. C

    Find the area of the loop using Green's Theorem

    Homework Statement Problem in attachment. Homework Equations The Attempt at a Solution Unfortunately I was unable to attend my only class where my proffessor taught this method of solving area. Plus my prof and classmates won't help me. Does anybody know how to solve area...
  43. M

    MHB Green's Theorem Verification for Vector Field F and Region R

    Hey! :o I have to verify the Green's Theorem $\oint_C{ \overrightarrow{F} \cdot \hat{n}}ds=\iint_R{\nabla \cdot \overrightarrow{F}}dA$. The following are given: $$\overrightarrow{F}=-y \hat{\imath}+x \hat{\jmath}$$ $$C: r=a \cos{t} \hat{ \imath}+a \sin{t} \hat{\jmath}, 0 \leq t \leq 2 \pi$$...
  44. M

    MHB Proving Stokes' Theorem with Green's Theorem

    Hey! :o $\overrightarrow{F}=M \hat{i}+ N \hat{j}+ P \hat{k}$ To prove the Stokes' Theorem we apply Green's Theroem at $ABE$, $BCE$, $CDE$. $(\oint_{ABE}+\oint_{BCE}+\oint_{CDE}){ \overrightarrow{F}}d \overrightarrow{R}=\iint_{ABCDE}{ \nabla \times \overrightarrow{F} \cdot \hat{n}}d \sigma$...
  45. J

    Green's theorem and area calculation

    In wiki there is the follows formula: https://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation But, I don't understand why M = x and L = -y. I don't found this step in anywhere.
  46. S

    How to Verify Green's Theorem for a Given Rectangle?

    Homework Statement Verify Green's Theorem in the plane for the \oint [(x^{2} - xy^{2})dx + (y^{3} + 2xy)dy] where C is a rectangle with vertices at (-1,-2), (1,-2), (1,1) and (-1,1). The Attempt at a Solution This means you have to use green's theorem to convert it into a double...
  47. T

    Green's Theorem: Explaining the Bounds Reversal

    I’m having a little trouble understanding why Green’s Theorem is defined as; ∮_C P dx+Q dy = ∬_D [(δQ/δx)-(δP/δy)] dA Instead of; ∮_C P dx+Q dy = ∬_D [(δQ/δx)+(δP/δy)] dA When proving the theorem, in the first step you simply reverse the bounds of the second integral to get the...
  48. A

    What am I doing wrong when solving this Green's theorem problem?

    Homework Statement Homework Equations The Attempt at a Solution $$Q=x\quad P=y^2-2y\\\oint_C{Pdx+Qdy}\\=\int_{C1}(y^2-2y)dx+xdy+\int_{C_2}(y^2-2y)dx+xdy\\=\int_{-\pi/2}^{\pi/2}(((sint+1)^2-2(sint+1))(-sint))dt+cost(cost)dt\\=\int_{-\pi/2}^{\pi/2}(2sin^2t+4sint+2)\\=3\pi$$ Correct answer is...
  49. M

    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. W

    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...