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Questions tagged [greens-theorem]

This tag is for questions about Green's theorem. Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.

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Find a simple and smooth curve $C$ such that $\displaystyle \int_C\vec{F}\cdot d\vec{r}$ gets its maximum value

I've been trying to solve this problem for a while, but for too long couldn't I continue my partial solution. I would be glad if you could shed some light on my solution. The task: Given the vector ...
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1answer
73 views

Find the curve of the maximum value of work done?

Suppose $C$ is a simple close curve (i.e. it doesn’t intersect itself) in the first quadrant. If $F = (y^2/2 + x^2y, -x^2 + 8x)$, find the curve that produces the maximum amount of work done by $F$. ...
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Green's Identities for tangential operators - How to derive this identity?

Source: http://www.diva-portal.org/smash/get/diva2:652933/fulltext01.pdf On page 11 it says: For tangential operators Green's formula becomes $$(\nabla_{\Sigma}\cdot w ,v)_{\Sigma}=(n_{\Gamma}\cdot ...
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15 views

Difference between (Variation of Parameters) and Green Function.

I'm confused with both of them. Variation of parameter and green function. I read on some papers that using green function on pde or ode is just like variation of parameter. But wait, in some videos, ...
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1answer
22 views

Using Green's theorem to compute integral on curve

Prove: $$\ \int_C (\sin x - y^2)dx +(x-y \tan^{-1}(y^2))dy = 2.4 $$ where $\ C $ is the curve from $\ (1,2) $ to $\ (-1,2) $ on $\ y = x^2 + 1 $ Using green's theorem $$\ \int \int_D (Q_x - P_y)dx ...
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2answers
36 views

Calculate line integral L: $y=sinx$, $y=0$, $0\le x \le \pi$

There is an example to calculate the line integral $\oint_{L}P(x,y)dx+Q(x,y)dy$ The contour $L$: $y=\sin x$, $y=0$, $0\le x \le \pi$ $P(x,y)=e^{x}y$, $Q(x,y)=e^{x}$ The calculation has to be ...
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1answer
26 views

Green's formula for the Laplacian defined in a neighborhood of the surface

Source: https://arxiv.org/pdf/1705.00069.pdf On page 4, it says that the surface Laplacian of a function $u$ (I will use different letters here) defined on a neighborhood of the boundary $\partial M$...
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15 views

Evaluate the line integral using two methods : directly and green

Consider the line integral question bounds i get how to find the parametize equations. what i dont get is how they find the upper and lower bounds for the integrals. so for c1= y=0 ; x =t ; 0 c3 ...
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1answer
20 views

Closed loops on nonconservative vector field

Consider the following (very simple) nonconservative vector field: $$V = (y, -x)$$ sketched in the following figure: Roughly speaking, the integral along the central red circle is obviously $\neq 0$...
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70 views

What is the result of the integration by parts of $\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega$?

$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded domain with Lipschitz continuous and piecewise smooth boundary $\Gamma:=\partial \Omega$, $...
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1answer
45 views

How to evaluate integral of (x - y)(dx + dy) with Green's Theorem?

I want to evaluate the integral $\int(x - y)(dx + dy)$ along curve C where C is the semicircular part of $x^2 + y^2 = 4$ above $y = x$ from $(-\sqrt2, -\sqrt2)$ to $(\sqrt2, \sqrt2)$ using Green's ...
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1answer
44 views

Evaluating $I = \iint_D (x+y)\, dy\,dx$ using Green's Theorem

Let $D$ be the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$. I want to evaluate the following integral $$I = \iint_D (x+y)\, dy\,dx$$ using two methods: by direct integration, and by ...
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1answer
32 views

Green's theorem over an annulus

I need help with this problem: Verify Green's Theorem in the plane where $S$ is the annulus $\{(x,y)\in\mathbb{R^2}|a^2\leq x^2+y^2\leq b^2\}$ and $F(x,y)=\left(\frac{-y}{\sqrt{x^2+y^2}},\...
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1answer
50 views

Green theorem intuition

What I have a hard time understanding is the connection between line integrals of vector fields and Greens theorem. It was explained that taking lines integrals of parametrized curves is to be ...
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1answer
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Is it “Valid” to prove Stokes' Theorem with Green's Theorem?

In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in ...
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1answer
73 views

Find a curve $\gamma$ satisfying $\int_\gamma y^3 \sin^2(x) \, dx - x^5 \cos^2(y) \, dy = 0$

Let a closed curve, $\gamma$, be parameterized by a function $f : [0, 1] → \mathbb{R}^2$ with a continuous derivative and f(0) = f(1). Suppose that $$ \int_\gamma y^3 \sin^2(x) \, dx - x^5 \cos^2(y) \,...
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1answer
20 views

What is a region of type 3 with regards to Green's Theorem?

I understand that a region of type 1 is where two curves are connected by two vertical lines and that a region of type 2 is where two curves are connected by two horizontal lines. But what is a region ...
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1answer
32 views

Path Integral equals zero on non conservative field

I was doing some excercises and I was asked to compute the line integral along certain path. I used greens formula to calculate the work. When computing the integral I had to divide the domain in two ...
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Greens function representation of nonlinear Poisson equation

Let $L$ be an operator and suppose the Green's function exists. That is there exist a function $G$ such that $LG=\delta$ where $\delta$ is the Dirac delta function. If $L$ is linear, one can represent ...
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2answers
50 views

Green's Theorem confusion

I have two equivalent forms of Green's theorem, namely $$ \int\int_D \frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}dxdy = \int_C pdx + qdy $$ $$ \int\int_D \frac{\partial p}{\partial x}+\...
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Green's theorem Area formula additional cases

In this thread, the OP states 3 area formulas and asks for proof. The answer given uses 3 specific cases where ∂Q/∂x − ∂P/∂y=1. I see why these 3 cases give the area formulas, but what about other ...
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1answer
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For Green's theorem, why is the region of integration of the line integral a weird partial derivative character?

Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem? $$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$ ...
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1answer
16 views

What does it mean for a region to be simultaneously a region of type 1 and type 2?

I am going through a proof of Green's Theorem for a simple region and I understand the mathematics taking place but do not understand the origins. 'Regions that are simultaneously of type I and II ...
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Looking for collection of exercises on Greene's theorem, Stokes theorem and the Divergence theorem

As the title states, I am looking for resources containging excersises on Greene's theorem, Stoke's theorem and the Divergence theorem. Ideally the excersises would be of computational nature (i.e. ...
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36 views

Integral on surface and boundary

Let $\Omega$ be connected bounded open set in $\mathbb{R}^{n}$. Let $U:\Omega\rightarrow \mathbb{R}^{n}$ be a $C^{1}$ vector field. The divergence theorem is given \begin{align} \int_{\Omega} \nabla\...
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Can someone explain to me the jump between steps in the bottom two lines of this proof (not yet totally complete)?

Suppose we have a region $G$ that is bounded by the straight lines $x=a$, $x=b$, $y=c$ and by an arc $y = f(x)$ (which lies above $y=c$) where $a \leq x \leq b$. If $f$, $P(x,y)$ and $Q(x,y)$ are all ...
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33 views

Show that $\int_{\Omega}\det(DF(x))dx = \det(M)\mathrm{area}(\Omega)$.

Let $\Omega$ be a bounded open in $\mathbb{R}^{2}$ such that $\partial \Omega$ is a $C^{1}$ curve, $F: \mathbb{R}^{2} \to \mathbb{R}^{2}$ a twice differentiable function. For $x = (x_{1},x_{2}) \in \...
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1answer
71 views

find the simple closed curve of $F(x,y) = (y^3-6y)i + (6x-x^3)j$ using Green's Theorem which will have the largest positive value

$F(x,y) = (y^3-6y)i + (6x-x^3)j$ a. Using Green's Theorem, find the simple closed curve C for which the integral $ ∳F \cdot dr $ (with positive orientation) will have the largest positive value. b....
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An intuitive explanation for Green theorem and Divergence theorem

As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize. I think I have found a pretty intuitive way to think about the ...
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2answers
47 views

Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$

Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$ my question is , after calculating the integral using green theorem ...
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37 views

Finding the maximum value of the integral $\int_{C}x^2y-2y^2-5y{dx} +(2xy-y^2x){dy}$

Find the maximum value of $\int_{C}x^2y-2y^2-5y{dx} +(2xy-y^2x){dy}$ , where C is closed curve with no self crossing taking in the positive direction. it is obvious that i need to calculate using ...
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3answers
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how to calculate $\int_{C}(2x^2+1)e^{x^2+y^2}dx+(2xy)e^{x^2+y^2}dy$ using Green theorem

Compute $\int_{C}(2x^2+1)e^{x^2+y^2}dx+(2xy)e^{x^2+y^2}dy$ where $C$ connects $(1,0)$ to $(0,1)$ by a straight line segment. I tried to use green theorem since $Q_x = P_y$ so $\int \vec F_\dot{}\vec ...
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how to calculate $\int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2}$ using green theorm or directly

Calculate $$\int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2},$$ where $C$ is the ellipse $3x^2 +5y^2 = 1$ taken in the positive direction. I tried to calculate the integral using green theorm. now i need to ...
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1answer
35 views

Integral of a differential form along a non-defined path

Let $R>0$ and $\Omega=\{(x,y)\in \mathbb{R^2}:x^2+y^2<R^2,y>0\}$. Consider also $\omega (x,y)=x^2dx+2xydy$. My goal is to prove that $\int_{\partial_{+}\Omega}\omega=\frac{4}{3}R^3$, where $\...
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1answer
37 views

Line integral in proof of Green's theorem

In wikipedia page about Green's theorem the following equality appears: $$ \int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx $$ I do not understand it. Wikipedia page about line integral defines ...
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1answer
68 views

Show that $F_x(x,y) = P(x,y), \ F_y(x,y) = Q(x,y)$.

In Complex Variables and Applications by Brown and Churchill it comes: When the point $(x_0,y_0)$ is kept fixed and $(x , y)$ is allowed to vary throughout a simply connected domain $D$, the ...
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Green’s theorem double into line integral

How I can convert this double integral into line $\int \int \frac{dx dy}{y^2} $ we have $\frac{dQ}{dx}-\frac{dP}{dy}=\frac{1}{y^2}$ how to find $Q, P$ functions? Any two functions works? $P=1, Q=\frac{...
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1answer
29 views

Trouble calculating line integral using Green's theorem, complicated integral.

"Calculate line integral of scalar function [$y(e^x) -1]dx + [e^x]dy$ over curve $C$, where $C$ is the semicircle through $(0, 10), (10, 0)$, and $(0, 10)$" I plan on using Green's theorem, and since ...
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1answer
119 views

Using Green's Theorem to find the flux

$F(x,y)=\langle y^2+e^x,x^2+e^y\rangle$. Using green's theorem in its circulation and flux forms, determine the flux and circulation of $F$ around the triangle $T$, where $T$ is the triangle with ...
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2answers
105 views

Verifying Green's theorem for a function

Let $G = \{ (x,y) \in \mathbb{R}^2 : x^2+4y^2 >1, x^2+y^2 < 4 \} $ $ \int_G x^2+y^2 d(x,y) $ I want to verify Green's Theorem : $ \oint_{ \partial G } f n ds = \int_G \operatorname{div}f\, ...
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1answer
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green theorem statement's meaning

Vector Calculus sector of 6.2 17. D is always on the left as we travel along C (C is the path of D) What is the meaning of the above statement? I can't understand ;(
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155 views

Area of a simple closed curve

Let C be a simple closed curve in a region where Green's Theorem holds. Show that the area of the region is: \begin{equation} A=\int_{C}xdy=-\int_{C}ydx \end{equation} Green's theorem for area ...
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31 views

Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field

Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field I have thought to do the following to solve this: ...
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1answer
54 views

Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ along a path

Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ when a particle moves from the origin, along the $x$ axis to the $(1,0)$, then on the line segment that joins the $(1,0)$ ...
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2answers
22 views

Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined

Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined by the inequalities $-a\leq x\leq a$, $-a\leq y\leq a$, ...
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1answer
43 views

Why is $ I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\zeta) d\zeta = -2i \int_{\partial \Omega} \frac{1}{z-\zeta}\partial u(\zeta) d\zeta. $?

I am unsure of some notation and also how a particular identity was derived. I read in a paper that because $\Delta = 4 \partial \bar \partial$ we have $$ I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\...
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2answers
69 views

Line Integral in second quadrant of Unit Circle

If I am asked to compute $$\int_c F . dr$$ Where $$F(x,y) = <d/dx f(x, y), d/dy f(x,y)>$$ and $$f(x,y) =\sin(x^3 + y^3)$$ and C is the portion of the unit circle in the second quadrant, ...
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1answer
24 views

Variational Formulation - inhomogeneous

I'm not sure how to get started with the following. Consider, $- \Delta u=f$ in $\Omega$ $u=u_o$ on $\Gamma$ I need to find a $u \in V(u_o)$ such that $a(u,v)=(f,v)$ $\forall v\in H^1_o$ where $...
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1answer
66 views

Green's Formula in the case where $M = x^2 - y^2$ and $N=2xy$

Question: Test Green's Formula in the case where $M = x^2 - y^2$ and $N=2xy$ and $\Omega$ is the triangle with vertices $(0, 0), (1,1), (2, 0)$. My attempt: $$\iint_\Omega \Big(\dfrac{\partial M}{\...
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1answer
141 views

Evaluate using Green's Theorem over ellipse

I have the answer to a problem and am trying to understand the steps to get to that answer. The problem is $\oint_C(x+2y)dx+(y-2x)dy$ around the ellipse C, defined by $x=4cos\theta, y=3sin\theta, 0\...