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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500 views

Green's theorem for piecewise smooth curves

Green's theorem is usually stated as follows: Let $U \subseteq \mathbb{R}^2$ be an open bounded set. Suppose its boundary $\partial U$ is the range of a closed, simple, piecewise $C^1$, positively ...
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2answers
143 views

Evaluate the integral F • dr along C

Evaluate the integral $F • dr$ along $C$ where $F$ is the vector function $F(x,y, z) = < -y^2, x, z^2 >$ and $C$ is the curve of the intersection of the plane $y + z = 2$ and the cylinder $x^2 ...
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1answer
570 views

Line integral using Green's theorem with a singularity in (x,y) = (0,0)

I'm trying to resolve this line integral who has a vector field with a singularity so with Green's theorem, using that the curl = 1 it's very difficult. My curve is $$C(t) = (3/2\cos(t) - \cos^2(t) , ...
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442 views

Closed line integral of conservative field not zero?

show that if $\mathbf{F}(x,y)=\frac{-y\mathbf{i}+x\mathbf{j}}{x^2+y^2}$, then $\oint\mathbf{F}\dot{}d\mathbf{r}=a\pi$ for every simple closed path that encloses the origin. Find the constant $a$. I ...
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1answer
147 views

Green's Theorem $F = (x - e^x cos y)i + (x + e^x sin y)j$; $C$ is the lobe of the lemniscate $r^2 = sin 2θ$ that lies in the first quadrant.

Using Greenʹs Theorem, compute the counterclockwise circulation of F around the closed curve C: $F = (x - e^x \cos y)\vec{i} + (x + e^x \sin y)\vec{j}$; $C$ is the lobe of the lemniscate $r^2 = \...
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1answer
153 views

An integral representation in the unit disk

I was experimenting with some complex analysis and i have some problems with my considerations. Consider $f$ holomorphic ,with $f : \Omega_1\to \Omega_2 ,$ where both $\Omega_1,\Omega_2$ are open ...
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1k views

(Multivariable) Green's Theorem Application

Well, I'm assuming it involves Green's Theorem. I'm not too sure actually. Let D be a region in the xy plane. Let A = ∬ dx dY (for region D). Let boundary of D be the region in which every point (x,...
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276 views

Using Green's Theorem to find area

Hi, I'm little confused to use green's theorem for the area with using C1 and C2 curves. I tried to find the area by using polar coordinates but I ended up with a diverging integral. I calculated ...
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2answers
514 views

Double Integral Formula for Holomorphic Function on the Unit Disc (Complex Plain)

While studying some complex analysis , I encountered the following problem: "Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$.Prove that for every $\zeta \in \mathbb{D}$ the ...
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83 views

Aproximation to Jordan curve by polygonal jordan curve

In the proof of the Green theorem (in the last step: any region can be approximated as closely as we want by a sum of rectangles), I need to prove the following result: Let $\gamma$ be a Jordan curve ...
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1answer
493 views

The area of the region enclosed by $C$ can be written $\frac{1}{2i}\int_C{\bar{z}} \ dz$

Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by $C$ can be written $$\frac{1}{2i}\int_C{\bar{z}} \ dz$$ Here's what I've done: $$\int_C{f(z)} \ ...
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81 views

Cauchy-Goursat's theorem and residue theorem

I am a little bit confused about using these theorems. For example, if I want to find the value of $\int_C e^{az}/z dz$ given that C is the unit circle with positive orientation, and a is a real ...
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2answers
153 views

Using Green's First Identity

Use Green's first identity to show that, $$\Delta_2u=u^3\qquad on\qquad x^2+y^2<1$$ $$u=0\qquad on\qquad x^2+y^2=1$$ has no twice continuously differentiable solution other than $u(x,y)$ ...
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88 views

Checking the validity of Green function using a operator using a certain limit

We have an operator of green function which can be written as $\Big(-i \frac{\partial}{\partial x} -a \Big)^2$ within the periodic boundary conditions, where $x,y \epsilon ]-\pi, \pi].$ Using ...
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214 views

Green's theorem application in Complex analysis

Let $\phi\in C_c^{\infty}(\mathbb{C}).$ Prove that $\displaystyle \int_{|z-w|>\epsilon} \log|z-w|\Delta\phi(z)dA(z)=\int_0^{2\pi}(\phi(w+re^{it})-r\log r\frac{\partial \phi}{\partial r}(w+re^{it})|...
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69 views

Verify Green's theorem for $ M = -y/(y^2 + x^2) $ , $ N = x/(y^2 + x^2) $

Verify Green's theorem for $$ M = -y/(y^2 + x^2) $$ $$ N = x/(y^2 + x^2) $$ $$ R = \{ (x,y) / h^2 \le x^2 + y^2 \le 1 \} $$ where $ 0 \lt h \le 1 $ My attempt: $ \int Mdx + Ndy = \int_0^{2\pi}...
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1answer
135 views

Area using Green Theorem [closed]

Please can anyone tell me how to prove that the area of a region enclosed by a simple closed curve $C$ is $$\frac 1 2 \int \limits _C x \Bbb dy - y \Bbb dx$$ using Green theorem? Thanks in advance.
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262 views

The idea behind Green's theorem [closed]

What is the intuition behind green's theorem? Does it relate an area with only its boundary? What does a line integral in green's theorem represent? Also, why is it useful? I prefer the most simple ...
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114 views

Green's theorem in vector analysis

I have tried working it out, but I don't seem to know how to use the y2=2x. the question is asking: Evaluate the integral $\int(x^2-2xy)dx + (xy^3+3) dy$ around the boundary of the region bounded by ...
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584 views

Verify Green's theorem for circle centred at (x,y)=(a,0)

Verify Green's theorem on the plane for the vector field$\ \mathbf F =3xy\mathbf i -x\mathbf j$ along the circle$\ c$ of radius$\ a$ centred at $\ (x,y)=(a,0)$ with counterclockwise direction. ...
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1answer
80 views

Prove that $\lim_{\epsilon \rightarrow 0}\int_{\partial B_\epsilon} (φ∇g · n − g∇φ · n) ds = 2πφ(0, 0)$

Suppose $φ : \mathbb{R^2}\rightarrow \mathbb{R}$ is any $C^1$ function and let $g:\mathbb{R^2}-\{(0,0)\}\rightarrow \mathbb{R}$ given by $g(x, y) := \ln\sqrt{x^2+y^2}$ Prove that $\lim_{\...
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36 views

Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$

Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$. I know that this is just using Green's first formula, but I'm having a ...
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5answers
187 views

Evaluate the contour integral (Most likely without Green's Theorem)

$\int_{c}\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ where $C$ is the triangle with vertices at $(5,5), (-5,5),$ and $(0,-5)$ traversed counterclockwise. (Hint: Be careful about the hypotheses of any ...
2
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1answer
482 views

Green's Theorem with change of variables

Evaluate $$\int_C F dr$$ $$ F =< x2, xy > $$ $$ C: x^2/4^2 + y^2/9 = 1 $$ With y ≥ 0 positively oriented. For the circle $$ u^2 + v^2 = 1 $$ $$ u=x/a $$ $$ v=y/b $$ $$ x=au $$ $$ y=bv $$ ...
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1answer
668 views

(Closed) Line integral of Conservative Field.

Suppose we have a conservative Field $ \vec F: D' \subseteq R^2 \rightarrow R^2$ where D is a set of points inside a closed curve (for example all the points inside a circle). Say we have subset of D',...
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1answer
250 views

Using Green's Theorem to Express the Integral $I=\int_C (Pdx+Qdy)$ as an expression of $I_i=\int _{C_i} (Pdx+Qdy)$

Let $p_1,...p_n$ be points in $\mathbb{R}^n$. Let $P(x,y), Q(x,y)$ be functions with continuous derivatives in $ D=\mathbb{R}^2\setminus\{p_1,...p_n\}$ such that $Q_x-P_y=1$ for all $(x,y)\in D$. For ...
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1answer
142 views

Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}$ through the line segment

Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find $\int_{C}{\bf{F}}\...
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1answer
218 views

Calculating Line Integral: $\int_{dA} (x^2 dx + y^2 dy)$

Normally we are given points, but this time it's different: $$\int_{dA} (x^2 dx + y^2 dy)$$ $A$ $=$ ${(x, y)}$ in $R^2;$ $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ , $ -1 \leq y \leq \cos(x)$ ...
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1answer
66 views

Green's Theorem for $ \int <2y^2 + \sqrt{1+x^5} , (5x-e^y)> dr $ where $ C: x^2+y^2=4 $.

Use Green's Theorem to evalutae $$ \int <2y^2 + \sqrt{1+x^5} , (5x-e^y)> dr $$ $$ C: x^2+y^2=4 $$ C is positively orientated $$ \int \int (dN/dx - dM/dy) dA $$ $$ = (5 - 4y) dA $$ $$ \...
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2answers
1k views

Using Green's Theorem to compute counterclockwise circulation

Using Green's Theorem, compute the counterclockwise circulation of $\mathbf F$ around the closed curve C. $$\mathbf F = (-y - e^y \cos x)\mathbf i + (y - e^y \sin x)\mathbf j$$ C is the right lobe of ...
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1answer
73 views

Prove $\oint_C\vec F \cdot \hat n\;ds=4\pi\,(q_1+…+q_n)$

Let $F:\Bbb R^2-\{p_1,p_2,\dots,p_n\} \to \Bbb R^2$, where $\{p_1,p_2,\dots,p_n\}\in \Bbb R^2$ be defined as $F(x)=\sum_{i=1}^n q_i \nabla\left(ln||x-p_i||^2\right)$ with $\{q_1,q_2,...,q_n\}\in \Bbb ...
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amusing applications of greens theorem

which application do you find most amusing and essential to the appreciation of Greens theorem suitable for a third semester calculus student ?
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49 views

Integral path between 2 points

so I need your help calculating the next inegral: Calculate the integral $$\int(10x^4-2xy^3)dx -3x^2y^2dy$$ at the path $$x^4-6xy^3=4y^2$$ between the points $O(0,0)$ to $A(2,1)$ please explain me ...
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1answer
258 views

Prove the integral is always imaginary

Show that if f is analytic on D and γ is a closed curve in the region then the integral $$\int \overline{f(z)}f'(z)$$ is purely imaginary. I think this problem would use some extension of cauchy ...
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1answer
988 views

Double integral of off centre circle.

I have the vector field $F = (3xy,-x)$ along the circle $c$ (counter clockwise) which has a radius $a$ and centre $(a,0)$. I want to try and apply Green's Theorem to this, where I obtain $\int\int(-1 ...
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0answers
37 views

Understanding the line integral $\oint_c \vec{F} \cdot \,d\vec{r}$

I'd like to check my understanding of the following integral ( and hopefully, in the process, provide a page where other students can come to understand it as something other than a visual stimulus ...
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1answer
91 views

Proving $\displaystyle\iint_{D}\!\!u\,\Delta u\,dA<0$ where $D$ is the closed, unit disc.

Hello again guys and gals! I'm stuck on a problem that I thought was going to be simple for me to prove, but I was wrong (yet again). I will try not to be so long-winded this time (see my previous ...
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0answers
178 views

Green's first identity and the calculus of variations

UPDATE: I was able to solve this problem using iterative integration by parts. However, I still cannot find how Green's first identity would apply here. Suppose I had a multiple integral over $p$-...
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1answer
181 views

Green's theorem area formula

I am assigned to calculate the area beneath the curve $y=x^2$ and above the $x$-axis using the formula $$A=\frac{1}{2}\int_C x\,dy\,-y\,dx$$ from $0\le x\le2$ while this seems simple to me I ...
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0answers
152 views

Green's theorem with unit normal and del operator

By appropriately choosing the functions P and Q in Green's theorem, show that $$\iint_R\nabla^2 \phi\,\mathrm{d}A =\int \frac{\partial \phi}{\partial n} \, \mathrm{d}s, $$ where $\frac{\...
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1answer
186 views

Using Green's Theorem to evaluate the area enclosed by a line and curve

Question Use Green's Theorem to evaluate the line integral $$\oint_{C} (x^2 + xy) dx + x y^2 dy $$ where C is the boundary region trapped by the line $ y = 2x $ and the curve $ y = -2x^2 $. ...
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1answer
7k views

What is a x-simple and y-simple region? [closed]

I have read the definition, but I do not quite understand what it means.
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1answer
52 views

How to set up and evaluate line integral for non trivial ellipse?

My task is as follows: $$\: \mathcal{C}:\textbf{r}(t) = \big(1 + 2\cos(t)\big)\:\hat{i} + \big(3\sin(t) - 2\big)\: \hat{j},\enspace t\in [0, 2\pi].$$ Calculate $\oint_\mathcal{C} \textbf{F}\:d\...
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2answers
65 views

Evaluating a given contour integral using Green's Theorem

Let $D$ be the region in $\mathbb{R}^2$ that contains the points $(x,y) : x^2 + y^2 \leq 1$ and $y \geq 0 $. Let $C$ be the curve enclosing $D$ oriented against the clock. Evaluate $$ \int\limits_C \...
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3answers
453 views

Prove that the area of a region is $\int _{C} x dy = -\int _C ydx$ using Green's Theorem

If $\mathbf{c}$ is a simple closed plane curve whose image bounds a region R, and which is traversed counterclockwise, then the area of R is $\int _{c} x dy = -\int _c ydx$, where x and y are the ...
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1answer
30 views

Vector Calculus, Line Integral Problem

I am having trouble with the following problem. I would like to see how to set up the problem and if there is any other tips I should use to solve similar problems. Thank you. Let $F(x,y) = (2x + 3y,...
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1answer
370 views

Greens first identity

Good afternoon all. Using the greens first identity I am finding weak formulation. Can someone check I did this correct with below mentioned greens first identity $$\int_U (\nabla \cdot \nabla \...
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1answer
133 views

Why is this tensor identity true?

I encountered the following claim in a book (Gockenbach: Understanding and Implementing the Finite Element Method), and I can't make sense of the equation. "... the reader to verify that if $\sigma$ ...
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2answers
265 views

References for the “extended” Green and Stokes' theorem.

I was watching these videos from MIT's series: Green, Stokes. And I didn't understand the justification: their "extended version" of the theorems. I looked up on google and couldn't find many ...
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1answer
88 views

How does Green's theorem apply here?

Let $D$ be the region delimited by $$\partial D: \begin{cases} C_1: x^2 + y^2 = 5^2\\ C_2:(x-2)^2+y^2= 1\\ C_3:(x+2)^2+y^2 = 1\\ C_4: x^2+(y-2)^2= 1\\ C_5: x^2+(y+2)^2= 1 \end{cases} $$ I've sketched ...