Green's theorem Definition and 125 Threads

Homework Statement Excuse my terminology, not sure what the actual translations are. Find a simple (no holes in it), closed, positively oriented, continuously differentiable curve T in the plane such that: \int_{T}(4y^3+y^2x-4y)dx + (8x +x^2y-x^3)dy is as big as possible, finally...

This is a copy of the book: http://i38.tinypic.com/20faqnc.jpg I know the derivation part, I just want to see whether I understand why the -ve sign of ##-\frac {\partial f}{\partial y}dA## in a more common sense way. From looking at the graph for type I region, ##g_2(x)## is above...

I am working on the derivation of Green's Theorem. I might have more question in later post. I am looking at this pdf file: http://www.math.psu.edu/roe/230H/slides_14nov.pdf In page 3, ##\int \int_R\frac{\partial N}{\partial x} dx dy=\int_c^d\int_a^{g(y)}\frac{\partial N}{\partial x} dx...

Here's the question: So using Green's Theorem, I got that the integral is equal to \int_{C}\frac{\partial}{\partial x}(-e^xsiny) - \frac{\partial}{\partial x}(e^xcosy)dxdy = 0. But surely the answer can't be 0? What am I doing wrong?

Here is the question: Here is a link to the question: Use Green's Theorem to calculate the work done by the force F? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Suppose that F = ∇f for some scalar potential function f(x, y) = 1/2(x2 + y2) Let C denote the positively oriented unit circle, parametrized by r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏. Compute the flux integral of \ointF\bulletN ds, where N is the outward unit normal to C.Homework...

http://img546.imageshack.us/img546/3171/integralbo.jpg For the above expression, I was told that it can be proven using Green's Theorem on the line integral on the RHS, however I can't seem the prove the equality. Note that $G$, $H$, $f$ are functions of $x_1$ and $x_2$. So I apply Green's...

Homework Statement Solve: ∫(-ydx+xdy)/(x2+y2) counterclockwise around x2+y2=4 Homework Equations Greens Theorem: ∫Pdx + Qdy = ∫∫(dQ/dx - dP/dy)dxdy The Attempt at a Solution Using Greens Theorem variables, I get that: P = -y/(x2+y2) and Q=x/(x2+y2) and thus dQ/dx =...

Hey all, I was working through some problems in my spare time when I realized that I wasn't so satisfied with my understanding of how to use Greens theorem with holes. Can someone refresh my memory? More specifically: Lets say I want to take the line integral in some vector field of a curve C...

Homework Statement Use Green's Theorem to evaluate \int_c(x^2ydx+xy^2dy), where c is the positively oriented circle, x^2+y^2=9 Homework Equations \int\int_R (\frac{\delta g}{\delta x}-\frac{\delta f}{\delta y})dAThe Attempt at a Solution I have found \frac{\delta g}{\delta x}-\frac{\delta...

Homework Statement Use Green's theorem to evaluate the line integral: ∫y3 dx + (x3 + 3xy2) dy where C is the path along the graph of y=x3 from (0,0) to (1,1) and from (1,1) to (0,0) along the graph of y=x. 2. The attempt at a solution I've completed two integrals for both paths (y=x3 &...

Homework Statement Green's Theorem to evaluate the line following line integral, oriented clockwise. ∫xydx+(x^2+x)dy, where C is the path though points (-1,0);(1,0);(0,1) Homework Equations Geen's theorem: ∫F°DS=∫∫ \frac{F_2}{δx}-\frac{F_1}{δy} The Attempt at a Solution What...

Homework Statement a) Evaluate the work done by the force field F(x, y) = (3y^(2) + x)i + 4x^(3)j over the curve r(t) = e^(t)i + e^(3t)j, tε[0, ln(2)]. b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that joins the points (1, 1) and (2, 8). c) Find the...

I am reading Etgen's Calculus: One and Several Variables section on Green's theorem. I was wondering if there is any direct application of this concept to physics or is it only used to calculate areas?

Homework Statement Let C be the boundary of the region bounded by the curves y=x^{2} and y=x. Assuming C is oriented counter clockwise, Use green's theorem to evaluate the following line integrals (a) \oint(6xy-y^2)dx and (b) \oint(6xy-y^2)dyHomework Equations The Attempt at a Solution...

Homework Statement Evaluate the following line integral ∫y^2 dx + x dy where C is the line segment joining the points (-5,-3) to (0,2) and is the arc of the parabola x= 4-y^2 Homework Equations Green's Theorem ∫ Mdx + Ndy = ∫∫ (∂N/∂x - ∂M/∂y ) dy dx The Attempt at a...

Homework Statement Calculate the area of the region within the hypocycloid x^{2/3}+y^{2/3}=a^{2/3} parameterized by x=acos^{3}t, y=asin^{3}t, 0\leqt\leq2\pi Homework Equations In the problem prior to this one, I showed that the line integral of \vec{F}=x\hat{j} around a closed curve in the...

I meant Line integral. Homework Statement I want to find the path integral of a vector function F over a closed path in Euclidean space with z = 0. Homework Equations The Attempt at a Solution I was wondering if it is allowed to first use Stoke's theorem and then Green's theorem. I would...

Homework Statement Use Green's Theorem to calculate \int F dr Homework Equations F(x,y)= (\sqrt x +y^3) i + (x^2+ \sqrt y) j where C is the arc of y=sin x from (0,0) to ( pi,0) followed by line from (pi,o) to (0,0). The Attempt at a Solution We have \int f dx + g dy = \int...

Homework Statement Use Green's Theorem to evaluate this line integral Homework Equations \int xe^{-2x}dx+(x^4+2x^2y^2)dy for the annulus 1 \le x^2+y^2 \le 4 The Attempt at a Solution \displaystyle \int_c f(x,y) dx + g(x,y)dy+ \int_s f(x,y) dx + g(x,y)dy = \int \int _D1 (G_x-G_y)...

Homework Statement Calculate the folowing directly and with greens theoremHomework Equations \int (x-y) dx + (x+y) dy C= x^2+y^2=4 The Attempt at a Solution Directly x= r cos \theta, y=r sin \theta, r^2=4, dx = -r sin \theta d \theta, dy= r cos \theta d \theta Substituting I get...

Homework Statement Using Green's theorem, evaluate: http://s2.ipicture.ru/uploads/20120117/6p57O2HO.jpg The attempt at a solution \frac{\partial P}{\partial y}=3x+2y \frac{\partial Q}{\partial x}=2y+10x \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=7x To do the integration, I'm...

Describing "D" is Green's Theorem Homework Statement Let F(x, y) = (tan−1(x))i+3xj. Find \int_C F • drwhere C is the boundary of the rectangle with vertices (0, 1), (1, 0), (3, 2), and (2, 3), traversed counterclockwise. The Attempt at a Solution I have Qx = 3 and Py = 0. Therefore Qx...

Homework Statement Use Green's theorem to find the integral ∫C (y^2dx+xdy) when C is the following curve (taken counterclockwise): the ellipse x^2/a^2 + y^2/b^2 =1.Homework Equations Green's theorem: ∫C Mdx+Ndy = ∫∫R (∂N/∂x-∂M/∂y)dA The Attempt at a Solution I tried parametrizing the ellipse as...

Question: Evaluate using Green's Theorem and sketch R. The question (excluding the sketch) and the attempted solution are on the attached image. I may have gotten the solution, but the numbers seem funny. Where did I go wrong?

Hi. I have a problem with this exercise. I wanted to verify the greens theorem for the vector field F(x,y)=(3x+2y,x-y) over the path \lambda[0,2\pi]\rightarrow{\mathbb{R}^2},\lambda(t)=(\cos t, \sin t) The Green theorem says: \displaystyle\int_{C^+}Pdx+Qdy=\displaystyle\int_{}\int_{D}\left...

Homework Statement Use Green’s theorem to find the integral \oint_{\gamma} \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy along two different curves γ: first where γ is the simple closed curve which goes along x = −y2 + 4 and x = 2, and second where γ is the square with vertices (−1, 0), (1, 0)...

Homework Statement \int_{C} (xy^{2}-3y)dx + x^{2}y dy G is finite region enclosed by: y=x^{2} y=4 C is boundary curve of G. Verify Green's Theorem by evaluating double integral and line integral. The attempt at a solution Q = x^{2}y dQ/dx = 2xy P = xy^{2}-3y dP/dy =...

Homework Statement F(x,y) = y i + (x2y + exp(y2)) j Curve C begin at point (0,0) go to point (pi, 0) along the straight line then go back to (0,0) along curve y=sin(x) Find circulation of F around C Homework Equations The Attempt at a Solution Curve part 1 Using Green theorem I got...

Homework Statement Verify Green's Theorem for F(x,y) = (2xy-x2) i + (x + y2) j and the region R which is bounded by the curves y = x2 and y2 = x Homework Equations \int CF dr = \int\intR (dF2/dx - dF1/dy) dxdy The Attempt at a Solution For \int CF dr , r(t) = x i + x2 j...

Homework Statement Show that for a solution w of Laplace's equation in a region R with boundary curve C and outer unit normal vector N, \int_{R}\left\| \nabla w\right\| dxdy = \oint_{C}w\frac{\partial w}{\partial N}dsHomework Equations The book goes through the steps to show that the following...

Homework Statement I have a doubt in proving the attached theorem. I have found that divergence theorem can be applied. However I am not able to arrive at the exact equation. Homework Equations Attached The Attempt at a Solution Derived the LHS. Couldn't proceed from there.

Homework Statement Use Green's Theorem to evaluate the line integralalone the given positvely oriented curve. ∫_{c} sin(y)dx+xcos(y)dy, C is the ellipse x2+xy+y2=4 Homework Equations The Attempt at a Solution ∫∫(cos(y)-cos(y))dA=∫∫0dA Because this ends up being the double...

Homework Statement Use Green's Theorem to evaluate ∫F*dr. (Check the orientation of the curve before you applying the theorem.) F(x,y)=<y2cos(x), x2+2ysin(x)> C is the triangle from (0,0) to (2,6) to (2,0) to (0,0) *=dot product Homework Equations Green's Theorem The Attempt...

Homework Statement \ointxydx+x^2dy C is the rectangle with vertices (0,0),(0,1),(3,0), and (3,1) Evaluate the integral by two methods: (a) directly and (b) using green's theorem. Homework EquationsThe Attempt at a Solution Evaluating the integral directly: c1: y=0,x=t,dx=dt,dy=o...

Homework Statement Use a line integral to find the area of the region enclosed by astroid x = acos3\phi y = asin3\phi 0 \leq \phi \leq 2\pi Homework Equations I used Green's Theorem: \oint_C xdy - ydx The Attempt at a Solution I solved for dx and dy from my parametric equations. I then...

The Integral I is defined by I = Integral F . dr Where F = (x-y, xy) << This is a verticle vector, i just didn'nt know how to write it with latex. And C is a triangle with the vertices (0,0), (1,0) and (1,3) tracked anticlockwise. Calculate the line integral using greens...

Homework Statement Find the simple closed integral of (x+xy-y)(dx+dy) counterclockwise around the path of straight line segments from the origin to (0,1) to (1,0) to the origin... a)as a line integral b)using green's theorem Homework Equations Eq of line segment r(t)=(1-t)r0+tr1 Greens...

Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...

Alright, I have a conceptual question regarding Green's Theorem that I'm hoping someone here can explain. We recently learned in my college class that, by Green's Theorem, if C is a positively-oriented, piecewise-smooth, simple closed curve in the plane and D is the region bounded by C, then the...

How to use green's theorem when the C is oriented clockwise

Homework Statement Use GT to find the area of one petal of the 8-leafed rose given by r=17sin(\theta) Recall that the area of a region D enclosed by a curve C can be found by A=1/2\int(xdy - ydx) I calculated it using the parametrization x=rcos(\theta), y=rcos(\theta) And I found a...

So let's say we have the vector field x^2yi+xy^2j, obviously the field is not conservative since dq/dx-dp/dy=y^2-x^2=/=0 however, let's say we wanted to find where locally the field would behave like a potential field, so we set y^2-x^2=0, so y=x (along the y=x line the field behaves like a...

Homework Statement Using Green's Theorem, (Integral over C) -y^2 dx + x^2 dy=____________ with C: x=cos t y=sin t (t from 0-->2pi) Homework Equations (Integral over C) Pdx + Qdy=(Double integral over D) ((partial of Q w.r.t. x)-(partial of P w.r.t. y))dxdyThe Attempt at a Solution I'm...

We can use Green's theorem to understand why the Exact Differential Equation satisfy the conditions it should have ... How about a DE for more than two variables ? Eg.dF=P(x,y,z,w)dx+Q(x,y,z,w)dy+R(x,y,z,w)dz+S(x,y,z,w)dw IF the equation above is an Exact Differential Equation , what...

any good notes/videos concerning green's theorem in plane? unfortunately missed my double lecture on it due to illness

Homework Statement Calculate the area of the region within x3 + y3 = 3xy. It can be parametrized by \gamma:[0,\infty] \rightarrow R2 with \gamma=<3t/1+t3, 3t2/1+t3>.Homework Equations Area = 1/2 \intx*dy - y*dx The Attempt at a Solution My plan is to take the curve parametrized by...

Homework Statement Use Green's theorem to compute the area of one petal of the 28-leafed rose defined by r = 5sin(14 \theta) Homework Equations A = \frac{1}{2} \int_c{x dy - y dx} \int \int_c{M_x + N_y}dx dy The Attempt at a Solution I'm really more confused about just what to do...

Homework Statement Solve: \oint x^{99}y^{100}dx + x^{100}y^{99}dy Assuming that it satisfies the conditions for Green's theroem, and: y = \sin{t} + 2, x = \cos{t}, 0 \leq t \leq 2\pi Homework Equations Green's theorem. The Attempt at a Solution \frac{\partial P}{\partial y} =...

Homework Statement Use Green's theorem to compute the area of one petal of the 8-leafed rose defined by r=9sin(4theta) It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as A =(1/2)int xdy-ydx. Homework Equations A =(1/2)int xdy-ydx The...