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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Why is Stokes theorem not applicable / not used correctly in this case?

I am doing some exercises and I don't understand what is wrong with my solution here. The problem is: given the integral $$ I = \int_S (1 + x^2) f(x) dydz - 2xy f(x) dz dx - 3z dx dy$$ Find such a ...
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Line Integral with absolute function

I'm trying to solve this line integral: $$\oint_{L} |x|dxdy$$ $$L: x^2 + y^2 = 1$$ So by dividing the domain into 2 half (left and right) and using Green's Theorem I can solve it. My question is when ...
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Greene's Theorem On Triangular Region [closed]

$\int_C xy^2\mathrm{d}x + 2x^2y\mathrm{d}y , $ C is the triangle with vertices (0,0), (2,2),(2,4) . My attempt : I drew the region And I'm taking orientation counterclockwise wise but now I'm not ...
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Balayage of delta measure at a point evaluated on fixed Borel set is a measurable function

I am trying to understand why if $G \subset \mathbb{C}$ is a bounded domain, and $\widehat{\mu}$ refers to the $\textbf{balayage}$ of the finite (positive) compactly supported measure (Borel measure) ...
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1 answer
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Reasoning behind Fields with "holes"

For this vector field $$F(x, y) = (-\frac{y}{(x+1)^2+y^2} - \frac{y}{(x-1)^2+y^2}, \frac{x+1}{(x+1)^2+y^2} + \frac{x-1}{(x-1)^2+y^2})$$ I'm asked to check if it is a gradient field in region $D = \{(x,...
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Fresnel-Kirchhoff’s Diffraction Formula Vs. Angular Spectrum Method

I tried to work through this lecture and first of all, I'm curious, what is the reason for this: We would like to use Green’s theorem with $v(x) = G(x − x0)$ As far as I understand, Green's theorem ...
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Work done by a field with singularities

The field is $F(x, y) = (\frac{-y}{(x-1)^2+y^2}, \frac{x-1}{(x-1)^2+y^2})$. And I want to calculate the work done along the circle $C_1:(x+1)^2+y^2=1$. I did verify that $\nabla \times F = \vec{0}$ so ...
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  • 477
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Taking Green's Theorem from 3D

I'm trying to understand the application for Green's Theorem for a field with singularities and taken from 3D to 2D point of view. So lets say I have this field $F(x, y, z) = (\frac{z}{x^2+y^2} + x, y,...
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  • 477
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Work done by a field along a circle

I have the vector field $$F(x, y) = (\frac{-y}{(x-1)^2+y^2} + \frac{y}{(x+1)^2+y^2}, \frac{x-1}{(x-1)^2+y^2} + \frac{-x-1}{(x+1)^2+y^2})$$ Using Green's Theorem I want to calculate the work done by $F$...
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Integral applying Green's Theorem

I have a vector field $\vec{F} = (-2y, x)$ and the curve $\{(x, y) : x^2+y^2 \leq 1; y > |x|\}$. Now I want to calculate the work done by the vector field along the curve that goes positively. So, ...
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  • 477
2 votes
1 answer
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Green's alternative formula for integration by parts

I'm currently implementing a method to solve usual elliptic problems where the classical form is the following: $$-\text{div}(k\nabla u) + \vec \beta \cdot \nabla u + \gamma u = f$$ Due to an ...
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How to find the path parametrization when calculating the circulation of a vector field?

So I have this vector field $$V(x,y)=(xy,x+y)$$ that I am calculating its circulation with 2 methods (Using a parametrization and using Green's theorem), The domain we're working on it is $${(x,y)\in ...
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Prove the following formula

Given $\phi(x):\mathbb{R}^3\to\mathbb{R},\phi\in C_0^2$ phi is twice differentiable and with compact support prove $$\phi(0)=\iiint_{\mathbb{R^3}}(\frac{-1}{4\pi|x|})\nabla^2\phi(x)d\Omega$$ my ...
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Line integral on shifted circle directly and using Green's theorem

I am trying to compute the line integral $\int_C \omega$, where $\omega=-y\sqrt{x^2+y^2}dx+x\sqrt{x^2+y^2}dy$ and $C$ is the circle $x^2+y^2=2x$ directly and using Green's Theorem. I have managed to ...
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Why is the line integral along the vertical segment zero [closed]

I am reading Theorem 11.10 : Green's Theorem for Plane Regions Bounded By Piecewise Smooth Jordan Curves. I have doubt regarding the last line in page 381, which says the integral along each vertical ...
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Using Green's theorem to calculate the area bounded by half of a cycloid

The area bounded by half of a cycloid $\alpha(t)=(R(t-\sin(t), R(1-\cos(t))$ where $R>0$ and $0 \leq t \leq \pi$ and the x-axis is: I've tried to use Green's theorem to solve this, by "...
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A problem of Green's theorem in a plane

Verify Green's Theorem in a plane for $$\ \int_C (\sin x - y^2)dx +(x-y^2)dy $$ where $\ C $ is the boundary curve of the region $R=\{(x,y)| y \ge x^2 + 1, y \leq2\}$. Here $P=\sin x - y^2$ and $Q=x-...
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Use Green's theorem to evaluate a line integral in a plane

The line integral is reduced to a double integral as $$\int\int_R 6x dxdy$$ But how to solve the double integral?
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2 votes
2 answers
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Green's theorem and translated angular form on the path $\gamma:[0,3\pi/2]\to\Bbb R^2,\gamma(t)=(t,\pi\cos t)$-strange result

I would like to compute the following integral $$\int_\gamma-\frac{y}{(x-\pi)^2+y^2}dx+\frac{x-\pi}{(x-\pi)^2+y^2}dy,$$ where $\gamma:\left[0,\frac{3\pi}2\right]\to\Bbb R^2,\gamma(t)=(t,\pi\cos t).$ ...
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4 votes
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Deriving Green's theorem

The reasoning leading to Green's theorem in my course makes a step I don't understand, with no justification. We have a function $P:R\to \mathbb{R}$ that has a partial derivative with respect to $y$ ...
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Derivation of Area theorem (conformal mapping).

While solving the Area theorem , i'm facing trouble in understanding the equation in these two black boxes, i know how they write $\displaystyle A=\frac{1}{2}\int_{c} R^2 d\phi$, but how they got its ...
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1 vote
1 answer
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Computing areas using Green's theorem

I want to compute the area of the surface $B$ with boundary parametrised by $$ \gamma(t)=\left(\begin{array}{c} \sin t \\ 4 \cos ^{2} t+\cos t \end{array}\right), \quad t \in[0,2 \pi] $$ ...
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Understanding classic Green's theorem

I was reading a book about Sobolev Spaces and to prove Grene's Theorem for weak derivatives they have used the following statement of Green's Theorem: Let $\omega$ be an bounded open subset of $\...
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1 vote
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Transforming every volume integral into a surface integral

Helmotz decomposition theorem says, on one hand, that every vector field $F$ sufficiently smooth can be decomposed into the sum of a solenoidal field $\nabla\times \bf A$ and a gradient field $\nabla \...
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Double integral of $-yf_x+xf_y$ on a rotating disc has zero value [duplicate]

Question: $u(x,y)=-yf_x+xf_y,f\in C^1(\mathbb{R^2})$ , $I(\alpha)=\iint_{D_\alpha}u(x,y)dxdy,D_\alpha\colon(x-2\cos\alpha)^2+(y-2\sin\alpha)^2\leq 1$ . Prove:$\exists \alpha,I(\alpha)=0$ . Attempt: ...
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Why in Green's theorem can we not simply integrate $y \mathrm{d} x$ instead of $y \mathrm{d} x - x \mathrm{d} y$?

According to the multidimensional Stoke's theorem, in order to evaluate the integral of a form $\omega$, I just have to find a one-form $\alpha$ so that $d\alpha=\omega$ and then use $\int_{\partial \...
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Find a simple, closed curve of a certain vector field

A little help is needed for the question below: Find a simple, closed curve in $\mathbb{R}^2$ so the vector field $$F(x,y) = (2y + 4y^3 + 2xy^3, -5x^3 + 3x + 3x^2y^2 )$$ will have maximum circulation....
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3 votes
1 answer
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Why is Green's theorem a special case of Stokes theorem?

I have already seen related questions and don't understand. Please help me. $\oint_C \mathbf{A} \cdot d\mathbf{r} = \iint_S (\nabla x \mathbf{A})\cdot \mathbf{n}$ dS Let $A \leq P,Q,0>$ Then $\...
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0 votes
1 answer
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Evaluate the line integral over the curve of intersection [closed]

Evaluate $\int_c \frac{y^2}{2}dx + zdy + xdz$, where $c$ is the curve of intersection of the plane $x+z = 1$ and ellipsoid $x^2+2y^2 + z^2 = 1$. The keyword "intersection" guided me to set ...
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1 vote
0 answers
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Green theorem and oriented ellipsis.

Green's theorem provides an elegant way to understand the connection between the ideas of line integrals around closed curves and double integrals over regions. In particular, we may use Green's ...
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Green function Stokes equation

So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force, where $\textbf{P}=\textbf{P}(x,y,z)$,...
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Limit cycles, simply and non-simply connected regions

I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a ...
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3 votes
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Formula for area between a curve and it's chord

Consider the curve $C':=(t, t^2)$ on the interval $t\in [0,1]$. A little calculation shows that the formula $\int \limits_{0}^{1} y \cdot dx = \int \limits _{0}^{1} t^2 \cdot dt$ gives an area of $1/3$...
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1 vote
2 answers
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Why can one use Greens theorem in this problem?

Find the lineintegral $$\oint F\bullet dr$$ given the vector function $$F(x,y)=(x^2-y^2-3x)i+e^{x/ \sqrt{y}}j$$ and the curve C being the boundary of the area in the first quadrant where $x,y\geq 0$ ...
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1 vote
1 answer
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Proving the generalized Cauchy integral theorem using Green's theorem

I was looking for proofs of the generalized Cauchy integral theorem: Theorem (Generalized Cauchy integral theorem): Let $\Omega\subset\mathbb{C}$ be an open set and $\gamma:[a,b]\to\Omega$ be a ...
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1 vote
0 answers
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Calculating a line integral in $2$ ways

I want to evaluate the line integral $$\oint_{C} (3y)dx+(2x)dy$$ where $C$ is the boundary of $ 0 \le x \le \pi, \enspace 0 \le y \le \sin{x} $. I found with Green's theorem that the result is $-2:$ $...
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2 votes
1 answer
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Inferring behavior of a function on the plane from curve integral

Let $P,Q:\mathbb{R}^2\rightarrow\mathbb{R}$ be two $C^1$ functions on the plane. Denote by $\Gamma$ the unit circle. $(P^2+Q^2)|_\Gamma>0$ and $\oint_\Gamma \frac{PdQ-QdP}{P^2+Q^2}\neq0$. Prove ...
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Curl of a Unit Circular Vector Field

(Sorry if that isn't the formal name for it) The vector field $F=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ has a curl of $0$, but when I calcuate the line integral $\int F\cdot dr$ over the unit circle, ...
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2 votes
1 answer
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Calculate the circulation of the vector field alone a parameterized circle (Stoke's Theorem...?)

Find the circulation of the following vector field $\vec{F}(x, y, z) = \langle \sin(x^2+z)-2yz, 2xz + \sin(y^2+z), \sin(x^2+y^2)\rangle$ along the circle $\vec{r}(t)=\langle\cos(t), \sin(t), 1\rangle$ ...
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Would the Greens theorem be applicable in the following closed regions in the Vortex Field?

The Vortex field is given by $\vec{F}$ = <$\dfrac{-y}{x^{2} + y^2}, \dfrac{x}{x^{2} + y^2}$> I understand that although $ curl$ $\vec{F} = 0$, the field is not conservative because $\vec{F}$ is ...
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137 views

Green's theorem applied over a parallelogram

Using Green's theorem, evaluate the line integral $$I=\int_Cx^2(x^2+y^2)dx+y(x^3+y^3)dy,$$ where $C$ is the parallelogram with vertices $(0,0), (1,0), (2,2), (1,2)$, traversed in that order. My ...
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1 answer
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Green's theorem to prove area of a simple closed curve

Let $S$ be the region enclosed by a piecewise smooth simple closed curve $C$ in the $xy-$plane. Use Green's theorem to show that the area of $S$ is $\frac{1}{2}\int_C xdy-ydx$, where $C$ is oriented ...
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2 votes
2 answers
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Why is it that we can define a function when Green's theorem is zero?

When it says that "this shows we can define a function...". Why is this? Why do we get this from greens theorem?. We have a previous theorem that says If $f:Ω→C$ is a continuous function in ...
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Cauchy's integral theorem and Green's Theorem clarification

I was reading this page on Wikipedia: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem In the very end, it says that "we therefore find that both integrands (and hence their integrals) ...
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Normal derivatives in proof or Carleman inequality.

To setup my question, I need to define some functions: $\Omega\subset\mathbb{R}^n$ is an open, bounded and non empty set, $Q:=\Omega\times(0,T)$, $\omega_0\subset\subset\Omega$ is an open non empty ...
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0 votes
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Evaluate Line integral with Green's Theorem

Calculate $$\displaystyle\oint_C(x\sin(e^y)+xy)dx+(\frac{x^2}{2}e^y\cos(e^y)+x^2y^3)dy$$ where $C$ is the polygon with vertices $(-1,0), (0,1), (1,1), (2,0), (0,-2) $ oriented counter clockwise. I ...
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Evaluating closed path integral $\oint_C(aydx +2xydy)$ without greens theorem

The path is a rectangle in the x-y plane with vertices $(0,0), (a,0), (0,b),(a,b)$. How would I do this without greens theorem? I tried to parametrise it but that didn't work as I don't have functions ...
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3 votes
1 answer
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If $\int_{\Omega} [u_t + (f(u))_x ] \phi\, dt \,dx =0 $ for all $ \phi \in C_0^\infty$ ,then it's true for all $ \phi \in C_0$

Prove that: If $$ \DeclareMathOperator{\Dm}{\operatorname{d\!}} \int\limits_{\Omega} [u_t + (f(u))_x ] \phi \Dm t \Dm x =0$$ for all $ \phi \in C_0^\infty(\Omega) $ , then it holds even for ...
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1 vote
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The symmetry of green's function in a Dirichlet problem

Consider the Green's problem $$\Delta G(x,y|s,t)=\delta(x-s)\delta(y-t)\ \ on\ \ R\\ G=0 \ \ on \ \ \partial R $$ Check if $G(x,y|s,t)=G(s,t|x,y)?$ Can not we say that since the differential operator ...
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Find the value of k for which the line integral depends only on the coordinates of the end points of C.

∫_C[(1+ky^2)/(1+xy)^2 dx+(1+kx^2)/(1+xy)^2 dy] . Find the value of k for which the line integral depends only on the coordinates of the end points of C. Hence, for this value of k, determine the ...
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