# 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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### How to deal with singularity and greens theorem

Suppose I have some flow integral \begin{align*} \int_{\partial B}^{} f \mathrm{~d}\mathbf{n} \end{align*} whereby $f$ has a singularity inside $B$ ($B$ is some open ball). Is there a way I can still ...
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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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### 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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### 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 ...
1 vote
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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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### 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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### 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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### 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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### Greens function with pole at infinity - Ahlfors Conformal Invariants chapter 2

In the textbook 'Conformal Invariants : Topics in Geometric Function Theory' page 25, there is the following formula (highlighted in yellow): I am very stuck on how Ahlfors manages to get formula (2-...
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### Deformation of Flux or Circulation Integrals.

I have recently learnt about Green's Stokes' and the Divergence Theorems. I read here: http://www.supermath.info/CalculusIIIvectorcalculus2011.pdf. On page 31, it describes a deformation you can make ...
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### Question regarding the intuition behind Green's theorem

I'm learning about Green's Theorem ,it simply says that work done to move say a boat along a closed loop is like a giant circulation and it must equal the sum of all circulation (curl) in all ...
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### Compute the area of the set $T\subset R^2$of the points lying within the trace of the closed curve $γ = γ_1 ∪ γ_2 ∪ γ_3$

Hi i found this exercise in the solved exercise session in my university portal and there was something there that i did not understand and i was hoping i could receive some answers here if possible. ...
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### Evaluate $\oint_{C} (\cos x e^x + y)dx$

Let $C$ be the curve parameterized by $r(t)=(5+2 \cos t, 2 \sin t)$ for $0≤t≤2π$. I'm trying to evaluate $\oint_{C} (\cos x e^x + y)dx$ over this region. I tried to use Green's Theorem to solve this:...
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### Proving the rectangle theorem using Green's theorem

Question 1 I want to prove that given $f:\mathbb{C}\to\mathbb{C}$ and rectangle $R$ with boundary $\gamma$ anti-clockwise, the line integral over $f$ would be $0$. I'm pretty sure with my way ...
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### Proof of the variant of Gaussian Surface Integral

Anyone has investigated the generalization of Gaussian Surface Integral (or Green Theorem, Stokes Theorem, Divergence theorem) like this? \begin{align} \frac12\iint_{S}(\mathbf{x}\times\mathbf{n})(\...
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### Green's Theorem Concepts: Circulation in R2

I am trying understand how circulation density arises in Green's theorem and I'd like to know if my line of thinking is on the right track. Here it goes :). Idea We know that if we have a vector field ...
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### Calculating a line integration

I need to calculate the constant alpha so that the integral be independent of the path we take. $$\int_C [x \ln{(x² + 1)}\;+\;(x² + 1)y]dx + \alpha[\frac{x³}{3} + x + \sin{y}]dy$$ Answer: $\alpha = 1$ ...
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### Prove that $\iint_A \operatorname{div} F\,dy\,dx=\int_a^bF\cdot N\,dt.$ [duplicate]

let $A$ be a region which is the interior of $C^1$ curve $C$ (oriented counterclockwise), such that $$C(t)=(g_1(t), g_2(t)).$$ and $N$ is normal to the curve i.e. $$N(t)=(g_2'(t), -g_1'(t)).$$ you can ...
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Let $A$ be a region in the plane such that $$a\le x\le b \text{ and }g_1(x)\le y\le g_2(x).$$ We can parametrize the two functions as follows $$\gamma_1(t)=(t, g_1(t)) \text{ and } \gamma_2(t)=(t,g_2(... • 2,932 1 vote 1 answer 73 views ### What's the Gauss-Greens-Theorem actually saying? Watching a video about harmonic functions I've seen the "Gauss-Greens-Theorem" written as:$$\int_{B(a,b)}\text{div}\,F\,\mathrm{dx\,dy} = \int_{\partial B(a,b)}F_x\,\mathrm{dy}-F_y\,\mathrm{...
I was asked the following question: Given the following vector fields $$G(x,y)=(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}),$$ $$H(x,y)=a G(x,y-1) + bG(x,y+1),$$ and the path defined by  r=\{\ (x,y) \...