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 $C$.

125 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
10
votes
1answer
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= \...
4
votes
0answers
129 views

Gaussian integral over a wedge of the complex plane

I'm trying to evaluate, if it is possible, a Gaussian integral over a wedge of the complex plane $$ I = \int_W d^2z \,e^{-|z|^2 + z^*a + za^*}, $$ ($d^2z = \frac{dz\,dz^*}{2} = dx\,dy = r\,dr\,d\...
4
votes
1answer
94 views

Looking for some intuition behind why the area enclosed by a simple closed curve $C$ can be obtained by computing $\frac{1}{2i}\int_C {\bar{z}} \ dz$.

By some manipulation and an application of Green's Theorem, I am able to show that $$Area = \frac{1}{2i}\int_C {\bar{z}} \ dz $$ To me, this seems to be an unexpected result. Is there some intuition ...
4
votes
0answers
191 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 $p$-...
3
votes
0answers
55 views

Green's Theorem with logarithm

Question: Using Green's Theorem, show that for a region in the complex plane $D$, with $z_0$ not in $D$, $$\iint_D \frac{1}{z_0-z} \,dx\,dy = \oint_{\partial D} \log(z_0-z)\,dy.$$ First of all, I'...
3
votes
0answers
87 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 ...
3
votes
0answers
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 ...
3
votes
0answers
218 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})|...
3
votes
1answer
91 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 ...
2
votes
0answers
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}...
2
votes
0answers
60 views

Green's Theorem Corollary

Context: Suppose $w(z,\bar{z})\in C^1$ and $D$ a simply connected region bounded by a sufficiently smooth curve $\Gamma$. Note that $w$ is complex but is not required to be analytic. Then, $$ \boxed{\...
2
votes
0answers
82 views

Green's Theorem in the Plane: Circulation Density

The following is from Chapter 16.4: Green's Theorem in the Plane, Thomas's Calculus, 14th Edition: Circulation rate around rectangle $\approx \left( \dfrac{\partial{N}}{\partial{x}} - \dfrac{\...
2
votes
0answers
44 views

Is this equality true? (cauchy integral + green's theorem)

Is this just TRUE? $$\oint_{\partial S} f(z) \, dz = i\iint_S \bigg[\frac{\partial{f}}{\partial{x}}+i \frac{\partial{f}}{\partial{y}}\bigg] \, dx \, dy$$ Because I'm using it without a proof, and I ...
2
votes
0answers
126 views

Calculating the area of cardioid with trisectrix with green's theorem: Will the area of the loop be added twice? See picture inside

I have a cardioid with a trisectrix, making a loop inside. This is what the cardioid looks like: . My question is the following, if we use Green's theorem to calculate the area of C, which is the ...
2
votes
0answers
38 views

Solve line integral using Green's thereom

Given $\oint_c x^2+y^2dx + 2xydy $ and C is the boundry R bounded by the graphs of $y=\sqrt{x}, y = 0, x = 4.$ Here is what I have so far : $$\oint_c x^2+y^2dx + 2xydy = \int\int_R (\frac{dn}{dx}-\...
2
votes
0answers
50 views

Line integrals using Green's theorem

I want to calculate these line integrals using green's formula :$\textit{a)}$ $\oint_C \overline{z}dz$ $\textit{b)}$ $\oint_C z^2dz$ in the following cases : $\textit{i)}$ $C = \{z\in \mathbb{C}\;/\;...
2
votes
0answers
19 views

Numetrical integration of complex exponent integral, need cheapest way

In need to calculate the integral $I = \iiint_\Omega \exp(i \vec{k} \cdot \vec{x} ) dV$ where $\Omega$ is a finite domain, bounded by piecewise-continuous continuous surfaces $P_i$, namely $\partial ...
2
votes
0answers
293 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 ...
2
votes
0answers
38 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 ...
2
votes
0answers
160 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{\...
2
votes
0answers
179 views

Green's theorem application

Problem Determine all circles $\mathcal C$ on $\mathbb R^2$ such that $$\int_{\mathcal C}-y^2dx+3xdy=6\pi$$ My attempt at a solution If I call $P(x,y)=-y^2$ and $Q(x,y)=3x$, then I can apply Green'...
2
votes
0answers
180 views

Prove using Green's theorem that the boundary value problem has at most one solution

Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\...
1
vote
0answers
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 ...
1
vote
0answers
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$(x,y)=(2xy-y + \sin x, e^{y^2}+x^2)$. ...
1
vote
0answers
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 ...
1
vote
1answer
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) \,...
1
vote
0answers
15 views

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 ...
1
vote
2answers
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}+\...
1
vote
0answers
18 views

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. ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
52 views

Solution of Dirichlet problem and the minimization of energy

Could someone explain why can we have $\frac{\partial}{\partial c_k}E(w)=0?$ Why the author can derive respect to constants $c_k$? Also I don't understand the rest of the solution, it's supposed to ...
1
vote
0answers
36 views

Computing the integral using Green's theorem

I have been doing an exercise problem in Multivariable calculus, namely I used Green's theorem. But there is something in the solution that I do not fully understand. 1)(Marked by the red rectangle):...
1
vote
0answers
35 views

Existence of crosscut partitions for Green's Theorem

I'm trying to formulate a totally rigorous proof of Green's Theorem, say on a set $S$ bounded by a piecewise smooth simple closed curve. This is not the most general statement, but let's stick with it ...
1
vote
2answers
408 views

Understanding Green's Theorem Proof

Going through the proof for Green's Theorem there is one step that I am not clear about. $$ \begin{eqnarray} \int_C M dx+Ndy &=& \iint_R\bigg(\frac{\partial N}{\partial x}-\frac{\partial M}{\...
1
vote
0answers
87 views

General Green's formula for a linear differential operator

Let $(M,g)$ be a compact, oriented Riemannian manifold with boundary. Let $E,F$ be two Hermitian vector bundles over $M$, and let $P:\Gamma (E) \to \Gamma (F) $ be a linear differential operator of ...
1
vote
2answers
250 views

Use Green's Theorem to Find the Area

Use Green's Theorem to find the area enclosed by: $$y=9-x^{2},y=8x, y=\frac{2}{5}x$$ (The area in Quadrant 1) In class we only did examples of this type of problem that were very simple (eg. ...
1
vote
0answers
111 views

Integrating difficult function over a polygon

I want to numerically integrate $$\iint_P e^{ik(y-y^\prime)^2}e^{ik(x-x^\prime)^2} \,dx\,dy$$ where $P$ is the region defined by a general $N$-gon with an arbitrary number of sides (a polygon has $N=...
1
vote
0answers
222 views

Finding a curve C that satisfies the hypnothesis of Green's Theorem for the following function.

So, i need to find a curve $\mathcal C$, with Green's Theorem, such as(this is a line integral, but i don't know how to insert with MathJax, sorry) $$\int \left(\frac{y^{3}}{12} - y\right)dx - \left(\...
1
vote
0answers
22 views

Green's theorem for a surface

I am looking at a demonstration of how Green's theorem for a planar surface comes from Stoke's theorem for a general surface, for a surface in the x-y plane. However I do not understand one of the ...
1
vote
0answers
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,...
1
vote
0answers
607 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. ...
1
vote
0answers
56 views

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 ?
1
vote
0answers
47 views

Is my way of thinking correct about Green's Theorem?

Consider the vector field $$F(x,y) = F_1(x,y)\mathbf{i} + F_2(x,y)\mathbf{j} = -\frac{y}{x^2 + y^2}\mathbf{i} + \frac{x}{x^2 + y^2}\mathbf{j}$$ Show that $\frac{dF_2}{dx}=\frac{dF_1}{dy}$ for ...
1
vote
1answer
39 views

Multivariable Calculus: Evaluating a Line Integral Using Green's Theorem With Single Differentiable

here's a question I'm working on. We have this integral over a region and we wish to use Green's Theorem to evaluate it. $\int_D x\ln(y) dx$ $D:1\leq x \leq 2, e^x \leq y \leq e^{x^2}$ Here's ...
1
vote
0answers
172 views

Vector field flux calculation

Consider the vector field: $$F=((y+1)e^{y^2}\sin(y^3),0)$$ and the curve consisting of the three line segments $A$, $B$ and $C$. Where $A$ goes between $(1,-1)$ and $(1,1)$. $B$ goes between $(1,1)...
1
vote
1answer
530 views

Using Green's Theorem to find area enclosed by curve

Use Green's theorem to calculate the area enclosed by the curve: $x^{2/3}+y^{2/3}=4$ Knowing that $A=\frac{1}{2}\int_c xdy-ydx$ I know that there are already some questions and answers on this ...
1
vote
0answers
674 views

green's theorem relating to calculating area

In this problem, I have to find the area of that blob. Pretty much I have to see if $\,N_x-M_y\,$ is equal to $1$. For the first choice, it is equal to $1$, yet the answer key says it is $4$?
1
vote
1answer
51 views

Verification of Green's thereom ( homework help)

Verify Green's thereom: $\oint_C (x^2 + y^2 +cos(x))dx +(x^2 +y^2 +sin(y))dy $ where C is the boundary of the semicircle" $${(x,y) \in R^2 :x^2 +y^2 \leq 4,x \geq 0 } $$ Solution: Please tell me ...
1
vote
0answers
61 views

Integrating using Green's Theorem

Evaluate $\int_{\gamma}xy\ dx $ where $\gamma$ is the boundary of the square with vertices $(0,0),(1,0),(1,1),(0,1)$. Now Green's Theorem says that $D$ be a bounded domain with piecewise ...