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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22 views

$\sum_{j=1}^n \psi(\lambda_j)=\dfrac{1}{2\pi} \int_{\mathbb{C}} \triangle \psi(z) \log|P(z) | \ dm(z)$

Let $$P(z)=\prod_{i=1}^{n}(z-\lambda_i) $$ $$\psi\in C_c^2(\mathbb{C}) $$ how do we get $$\sum_{j=1}^n \psi(\lambda_j)=\dfrac{1}{2\pi} \int_{\mathbb{C}} \triangle \psi(z) \log|P(z) | \ dm(z)$$where $\...
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14 views

2-dimensional Divergence Theorem proof

I want to proof: $$\iint_D\nabla\cdot\mathbf G\,dA=\oint_{\delta D}\mathbf G\cdot\mathbf N\,ds$$ So far: $$\mathbf G(r)= \begin{pmatrix} F_2(r) \\ -F_1(r)\\ \end{pmatrix}$$ $$\...
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16 views

Integrate by parts the product of the divergences of two vectors. Green's Formula

I am integrating by parts $\int\limits_\Omega (\nabla \cdot \vec{u}) (\nabla \cdot \vec{v}) dx $ I found a general integration by parts formula in a book Mécanique des milieux continus et discrets ...
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15 views

Question about the application of Green's Theorem

Let $f$ be a positive and continuously differentiable function on $(-1,1)$ such that $f(-1)=f(1)=0$ and the area between $f$ and the $x$-axis is $\pi/4$. If $C=\text{graph}(f)$ is contained in the ...
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11 views

Doubts re: applying Green's Theorem to compute line integrals

Let $C_1$ be the plane curve defined in polar coordinates as $r(\theta)=\theta, 0 \leq \theta \leq \pi$, and F$: \mathbb{R}^2 \to \mathbb{R}^2$ the vector field F$(x,y)=(2xy-y + \sin x, e^{y^2}+x^2)$. ...
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38 views

Find the maximum of $\int_{\gamma} (x^2y-2y)dx + (2x-xy^2)dy$, and find the curve $\gamma$ that gives it.

The task is to find the maximum of $$\int_{\gamma} (x^2y-2y)dx + (2x-xy^2)dy$$ when $\gamma$ is a smooth, regular, closed and simple curve in $\mathbb{R}^2$, and $\gamma = \partial G$ for a bounded ...
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42 views

Conservative fields - Definition confusion [duplicate]

I have seen several definitions of conservative fields, where we always assume that the domain of a vector field $ \pmb F $ is simply connected before giving the definition of a conservative field. ...
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84 views

Can I apply the gradient theorem for a field with not simply connected domain?

Let $ \pmb G $ be a vector field with domain $ U \subseteq \mathbb{R^2}. $ If $ U $ is not simply connected, but there exists a function $ f $ such that $ \pmb G = \pmb \nabla f \; \; \forall \; (x,y)...
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119 views

Not sure about an alternative way of calculating a line integral

Calculate the line integral $ \oint_c \vec F \: d \vec s $ , where $$ \vec F(x,y) = \Biggl( \frac{\partial}{\partial x} \biggl( \frac{x}{x^2+y^2} \biggr)+1 \:, \; \frac{\partial}{\partial y} \biggl(...
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51 views

Using Green's Theorem on an non-closed curve by adding or subtracting another curve/line?

I understand Green's Theorem can only be used on curves that are simple and closed. However, taking a look at these two examples, it seems like you can add a line so that the curve becomes closed so ...
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189 views

Trying to compute limit of singular integrals : $L= \lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y.$

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
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41 views

Line integral with Green's theorem

$C$ is a closed curve, $(0,0)$ is surrounded by $C$, Let $X = ax+by,\quad Y= cx+dy,\quad ad-bc=-7$, compute integral $$ \int_C \frac{X \; dY-Y \; dX}{X^2+Y^2} $$ One of my idea is $$d(\arctan\frac{X}...
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42 views

Green's theorem Evaluation [closed]

I am tasked with solving the following question: Evaluate $\oint(x^2-2xy)dx+(x^2y+3)dy$ around the boundary of the region given by $y^2=8x$ and $x=2$ two ways: (a) Directly (b) Using ...
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71 views

How can Green's Theorem be used to derive Maxwell's equations?

I've learned how to prove Green's Theorem and I read that it contributed to deriving Maxwell's equations. How can Green's Theorem be used to derive any of four Maxwell's equations? What else do I have ...
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58 views

Help for evaluating a line integral with Green's theorem

I have the following line integral of kind 2 $$\iint (2x)dx+3(yx)dy$$ and the region $$C:4\cos(2t) \ , \ y=3\sin(2t)$$ I sketch the region and its an elipse: $$\frac{x^2}{4}+\frac{y^2}{3}=1$$i am ...
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32 views

Green's theorem with absolute value boundry

I need to apply green's theorem with the field $F(x,y)= (x+y,x-y)$ on a positive oriented region bounded between the circle $x^2+y^2=9$ and $|x|+|y|=1$, but when I try to parametrize the boundry, I ...
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65 views

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
97 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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25 views

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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42 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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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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42 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
30 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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17 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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79 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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48 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
46 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
44 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
69 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
60 views

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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74 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
24 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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34 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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54 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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20 views

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
17 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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46 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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35 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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76 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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58 views

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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50 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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40 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
42 views

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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2answers
64 views

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
36 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 $\...