What is Greens theorem: Definition and 29 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. 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...
  2. E

    Determining between direct evaluation or vector theorems

    So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...
  3. 159753x

    Simple divergence/Green's theorem question

    I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method. The problem: For one example, I am trying to find the divergence of some vector field from a...
  4. J

    Greens theorem boundary of a rectangle

    Homework Statement ##\mathscr{C}: x=1,x=3,y=2,y=3## ##\int_\mathscr{C} (xy^2-y^3)dx+(-5x^2+y^3)dy## Homework Equations The Attempt at a Solution ##\frac{\partial Q}{\partial x} = -10x^2 \,\,; \frac{\partial P}{\partial y} = 2xy-3y^2## ##\int\int_\mathscr{C} \frac{\partial...
  5. M

    Line integral and greens theorem

    Homework Statement \int \vec{F} \cdot d\vec{r} where F=<y,0> and \vec{r}=unit circle. Homework Equations i'd prefer to do this one without greens theorem (using it is very easy). The Attempt at a Solution y=r\sin\theta and x=r\cos\theta. now \int \vec{F} \cdot d\vec{r}=\int...
  6. J

    Verify Greens theorem half done

    Homework Statement Verify Greens theorem for the line integral ∫c xydx + x^2 dy where C is the triangle with vertices (0,0) (1,1) (2,0). This means show both sides of the theorem are the same. Homework Equations ∫c <P,Q> dr = ∫∫dQ/dx -dP/dy dA ∫c xydx + x^2dy The Attempt at a...
  7. S

    Greens theorem and parametrization

    Hello. I just wonder if anybody know if there are any rules, when to use parametrization to greens theorem in a vector line integral over a plane. Becouse, it seems sometimes, you have to parametrizice, and other places you dont. I get confused.
  8. G

    Greens Theorem Area of ellipse

    Homework Statement Find the area swept out by the line from the origin to the ellipse x=acos(t) y=asin(t) as t varies from 0 to t_{0} where t_{0} is a constant between 0 and 2\pi Homework Equations The Attempt at a Solution so using Greens Theorem in reverse i get A=\frac{1}{2}\oint_{c}...
  9. G

    Calculating a Line Integral with Greens Theorem

    Homework Statement Calculate \oint _c xdy-ydx where C is the straight line segment from (x_1 , y_1) to (x_2 , y_2) Homework Equations The Attempt at a Solution so from Greens theorem I get \oint _c xdy-ydx = \int\int \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} dxdy =2...
  10. H

    Understanding Greens Theorem: Exploring P and Q in Vector Calculus

    I understand Greens Theorem, been doing a bit of it recently, but I have perhaps a very... novice question. What is P and Q exactly? They showed us on the board, but I am unsure what they are. Are they vectors? Or are they functions of a vector?
  11. P

    Line Integral does not match Greens Theorem?

    Homework Statement To evaluate the following line integral where the curve C is given by the boundary of the square 0 < x < 2 and 0 < y < 2 (In the anti clockwise sense): \oint (x+y)^2 dx + (x-y)^2 dy The Attempt at a Solution Firstly it is noted that for a square ABDE : Between...
  12. K

    Question regarding Greens theorem

    Homework Statement I have some questions similar to this one. I have to just provide reasoning as to why this can or cannot be evaluated using greens theorem. given f = x/sqrt(x^2+y^2) dx + y/sqrt(x^2+y^2) dy, and the curve c is the unit circle around the origin. Why can/cannot the integral...
  13. B

    Calculating Area of Curve Using Greens Theorem

    Homework Statement Find area of curve using area formula of Greens theorem Homework Equations r(t)=(t-sin t) i +(1- cos t ) j for 0 <= t <= 2 pi. The curve is y = sin x The Attempt at a Solution Do i let x(t)=t...?
  14. B

    Greens Theorem: Evaluate Line Integral of 6xy-y^2

    Homework Statement Use greens theorem to evaluate this line integral \oint_{C} 6xy- y^2 assuming C is oriented counter clockwise. The region bounded by the curves y=x^2 and y=x. Homework Equations \displaystyle \int\int_{R}\Bigl( \frac{\partial Q}{\partial x} - \frac{\partial...
  15. B

    Greens Theorem: Evaluating Integral of x^2 y dx +(y+x y^2)dy

    Homework Statement Use greens theorem to evaluate the integral Homework Equations \int x^2 y dx +(y+x y^2)dy where c is the boundary of the region enclosed by y=x^2 and x=y^2. The Attempt at a Solution The integral is \displaystyle \int_{0}^{1} \int_{x^2}^{\sqrt {x}} y^2+x^2...
  16. L

    Using Green's Theorem to Solve a Circle Line Integral

    Homework Statement Use greens theorem to solve the closed curve line integral: \oint(ydx-xdy) The curve is a circle with its center at origin with a radius of 1. Homework Equations x^2 + y^2 = 1 The Attempt at a Solution Greens theorem states that: Given F=[P,Q]=[y, -x]=yi-xj...
  17. I

    Evaluating Int. on V Along C Using Greens Theorem

    Homework Statement V = (3y^2 - sin(x)) i +(6xy+√(y^4+1))j along teh closed path C defined by x^2 + y^2 =1, counterclockwise direction Homework Equations Greens Thoerem The Attempt at a Solution I am stuck on the limits part of the integration. I get so far into greens theorem to...
  18. D

    Damped Harmonic Oscillator Using Greens Theorem

    Hi there, I just started an intermediate classical mechanics course at university and was smacked upside the head with this question that I don't know how to even start. Homework Statement We are to find the response function of a damped harmonic oscillator given a Forcing function. The...
  19. N

    Greens theorem and geometric form

    <y-ln(x^2+y^2),2arctan(y/x)> region : (x-2)^2+(y-3)^2=1 counter clockwise taking int int dQ/dx - dP/dy dA leads to -int int dA here my text is showing the next step as a solution of -pi not sure ..polar cords ext..
  20. L

    What is the result of applying Greens theorem to these vector fields?

    Homework Statement Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine F for the following vector fields whether the line integrals int(Fdr) are positive negative or zero F = xi + yj F = -yi + xj F = yi -xj F= i +...
  21. D

    Surface Integral (and Greens theorem) confusion

    Hello all, Evaluate \int\int r. da over the whole surface of the cylinder bound by x^{2} + y^{2} = 1, z=0 and z=3. \vec{r} = x \hat{x} + y \hat{y} + z \hat{z} Sorry for the awkward formatting, this site is giving me trouble. Anyways, it seems to me that since I have 3 dimensions...
  22. S

    Greens theorem direction of line integral

    My course notes said that in greens theorem where the closed line integral of F.r = the double integral (...)dxdy the curve c is taken once anti-clockwise, why does it matter which way you take the line integral? Does it matter at all? Thanks
  23. T

    Greens Theorem for negatively orientated curve

    Hey All, in vector calculus we learned that greens theorem can be used to solve path integrals which have positive orientation. Can you use greens theorem if you have negative orientation by 'pretending' your path has positive orientated and then just negating your answer ? Regards...
  24. O

    Is my proof of greens theorem correct?

    here it is. http://img193.imageshack.us/img193/79/greenproof.jpg
  25. B

    Calculating Integral Using Greens Theorem

    Homework Statement Use greens theorem to calculate. \int_{c}(e^{x}+y^{2})dx+(e^{x}+y^{2})dy Where c is the region between y=x2y=x Homework Equations Greens Theorem \int_{c}f(x.y)dx+g(x,y)dy= \int_{R}\int (\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y})dA The...
  26. J

    Green's Theorem for Evaluating Line Integrals on Curves

    Homework Statement Evaluate the \int xyi - xj dot dr over the curve y = 1 - x2 from 1,0 to 0,1 Homework Equations The Attempt at a Solution I used greens theorem \int -1 - x dy dx dy is from 0 to 1-x2 dx is from 0 to 1 \int-x3 + x2 - x - 1 dx =...
  27. T

    Help with line integrals and greens theorem

    I get an answer for this problem, but its 0 and i think that's wrong. if someone could please, help that'd be great. Homework Statement Find the work using the Line Integral Method: W = Integral of ( Vector F * dr) Vector Field: F(x,y) = (xy^2)i + (3yx^2)j C: semi circular...
  28. H

    Greens theorem and cauchy theorem help

    I'm doing these in order to prepare for my quiz in a week. I have no clue where to get started or the first step in attempting problem 3 and problem 4. Please do not solve it, I just want a guide and a direction... thanks if you guys don't mind, please download and have a look!
  29. S

    Verifying Greens Theorem for Region R with P(x,y)=xy, Q(x,y)=x^2

    how can greens theorem be verified for the region R defined by (x^2 + y^2 \leq 1), (x + y \geq 0), (x - y \geq 0) ... P(x,y) = xy, Q(x,y) = x^2 > okay i know \int_C Pdx + Qdy = \int\int \left(\frac{dQ}{dx} - \frac{dp}{dy}\right) dA so: \int_C xy dx + x^2dy = \int\int_D \left(2x -...