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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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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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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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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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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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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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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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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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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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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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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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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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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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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1 vote
1 answer
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Line integral with Green theorem

I will compute $\int_C \ e^xdx+xydy$ where C is the triangle with vertices (0,0), (1,1) and (0,2) with a positive orientation. I started with $\iint (\frac{\partial Q}{\partial x}-\frac{\partial P}{\...
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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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1 answer
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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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1 answer
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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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1 answer
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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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1 answer
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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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1 answer
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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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1 vote
1 answer
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Calc $P=\int_C \frac{e^{-x^2y}}{\sqrt{( 1+x^2y^4 )^3}}( xy[ 2( 1+x^2y^4 )-y^3 ]dx+x^2[ e^{-x^2y}\sqrt{(1+x^2y^4)^3}+1+x^2y^4-2y^3]dy)$

Calc the integral: $$P=\int \limits_C \dfrac{e^{-x^2y}}{\sqrt{\left( 1+x^2y^4 \right)^3}}\Bigg( xy\left[ 2\left( 1+x^2y^4 \right)-y^3 \right]dx+x^2\left[ e^{-x^2y}\sqrt{\left(1+x^2y^4\right)^3}+1+x^...
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A line integral/Green's Theorem problem

I'm stuck on the following problem: Let $f(x,y) = -\frac{y}{x^2 + y^2}$ and $g(x,y) = \frac{x}{x^2 + y^2}$ for all $(x,y) \neq (0,0)$. Show that $$\oint_{\partial S} (f(x,y) \,dx + g(x,y)\,dy) = 2\pi $...
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In a line integral over region R enclosing a singularity, how to calculate integral over the singularity without using a limit (or is limit implicit)?

I have a question about a particular portion of my multivariable calculus book in the chapter on Green's Theorem. Let me first set the theory up. If you want to jump to the question it is further ...
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1 vote
1 answer
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Compute the following line integral: $\oint_C \frac{\text{d}x}{y} -\frac{\text{d}y}{x}$ Where $C$ is the circle $x^2+y^2=1$.

Compute the following line integral: $$\oint_C \frac{\text{d}x}{y} -\frac{\text{d}y}{x}$$ Where $C$ is the circle $x^2+y^2=1$. The function $1/y$ and it's first partial derivative with respective to $...
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2 votes
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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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2 votes
2 answers
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Compute $\oint\limits_{\partial\Sigma} \frac{xdy-ydx}{x^{2}+y^{2}}\,$

Assume $ \partial\Sigma$ is a positively oriented, piecewise smooth, simple closed curve in a plane and let $ \Sigma$ be the region bounded by $ \partial\Sigma$ containing $(0,0)$, then compute $$\...
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Problems while doing a line integral

let $C$ be a closed $C^1$path (oriented counterclock wise) consisting of a piece of $y^2=2(x+2)$ and the vertical segment $x=2$, and $F$ is a vector field such that $$F(x,y)=\left(\frac{-y}{x^2+y^2}, \...
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1 vote
1 answer
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Proving a special case of Green’s theorem

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(...
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1 vote
1 answer
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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{...
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Conditions needed to apply Green's theorem and "homotopic" argument

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) \...
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