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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Line integral where $C$ is the boundary of the square -Green's Theorem

I am trying to solve the following problem but I face difficulties with the results. Any suggestion? The line integral $\int_C y^2 dx - x dy$, where $C$ is the boundary of the square $[-1,1]\times[-1,...
Athanasios Paraskevopoulos's user avatar
5 votes
0 answers
71 views

Geometrical meaning of calculating area using Green's theorem

Green's theorem says that: $$ \int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy $$ If the M and N statisfy $\frac{\partial M}{\partial x} -...
Wojak2121's user avatar
1 vote
1 answer
54 views

The work done by the force $ \vec{F} $ on a particle

Question The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first ...
Gajjze's user avatar
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2 answers
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Could I use Green's Theorem here?

I want to solve for the line integral: $$\tag{1}\oint \alpha\nabla \phi_i\cdot \hat{\textbf{n}} ds$$ on the square boundary: $(0\le x \le 1, 0), (1,0 \le y \le 1), (1 \le x \le 0, 1),(0,1\le y \le 0)$ ...
Researcher R's user avatar
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1 answer
59 views

Confusion over why Green's Theorem is usable in one situation with closed circle and not the other.

As I understand it, if a region contains the origin, Green's Theorem cannot be applied. However, one question in my issued lecture notes appears to contradict this and another one appears to follow ...
Lim Min Kang's user avatar
0 votes
0 answers
31 views

Caculate the line integral of a ellipse by a small circle.

Calculate the line integral $$ \int_\gamma \frac{y\,dx+(1-x)\,dy}{(x-1)^2+y^2} $$ where $\gamma$ is the ellipse $x^2 + 4y^2 = 4$ traversed two laps in positive direction. So I have been given a ...
per persson's user avatar
1 vote
1 answer
33 views

Green's theorem and identities.

Using Green's formula we will prove that $\int\limits_{\partial D} \frac{\partial u}{\partial n} \,dS = \iint\limits_D \Delta u \,dx\,dy.$ Let $D$ be the region for which Green's theorem holds and $u: ...
Paull's user avatar
  • 65
1 vote
2 answers
129 views

Finding the area by Green's Theorem

If I have $\ x=\sin^3t$ and $\ y = \cos^3t$ I'm using Green's theorem to calculate the area enclosed by the curve via integration, but at the end I'm getting negative area $-3\pi/8$, which is wrong. ...
Mohaboko's user avatar
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Application of Green's Theorem on a polar closed curve C.

Question: Let $C$ be closed curve as the boundary of region $R$. $C$ is defined as the polar coordinate inequalities $1\le r\le2, 0\le t\le \pi$. Define the field $F(x,y)=P(x,y)i+Q(x,y)j$ where $P=x^2+...
user1259172's user avatar
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Understanding Conservation in Divergence Theorem

Ok so I'm struggling with the concept of conservation in the divergence theorem. Divergence theorem states that: $$ \iiint_O {\nabla \cdot {\bf F}}\ dV = \iint_{S=\partial O} ({\bf F}\cdot \hat{{\bf n}...
Sterling Butters's user avatar
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1 answer
50 views

Bounded second derivative also bounds the function

I've been struggling with this problem for a few days: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=xg(y)-yg(x)$, where $g:\mathbb{R}\rightarrow \mathbb{R}$ is such that $g\in C^2$, $...
Arthur's user avatar
  • 35
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1 answer
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Is the Normal vector is pointing inside or outside?

Good morning, I am posting this again because I am having trouble to understand this post (How do I check if the normal vector is pointing inside or outside?). Here is my question : I have the ...
Ravinala's user avatar
1 vote
0 answers
45 views

How to transform area integral $\int_{D} \omega^2 \ dx \ dy$ into boundary integral $\oint_{C} \square \ ds$?

Let $\omega$ be a function that satisfies the Laplace's equation $$\nabla^2 \omega = 0$$ The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the ...
Carlos Adir's user avatar
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Stokes / Green's theorem with non-regular regions

One statement of Green's Theorem (Stewart) I have seen is: Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$. If $\mathbf{F}...
Alex B's user avatar
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Line integral exercise of a real vector field

I'm trying to solve the following exercise of a vector field over line integral: $$\int\limits_C\frac{-y}{4x^2+9y^2}dx+\frac{x}{4x^2+9y^2}dy,$$ where $C$ is the closed curve formed by the equations $y=...
Fernando's user avatar
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Weaker version of Goursats theroem

I have a homework problem that states: Let $\Omega \subseteq \mathbb{C}$ open. Suppose that $f:\Omega \rightarrow \mathbb{C}$ is holomorphic and $C^1$. Show that: $\int_{T}f dz = 0$ Where $T$ is an ...
strugglingStudent's user avatar
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63 views

Green's theorem example application for engineers and physicists.

I am looking for example applications of Green's theorem (in $2D$) that appeal to physicists or engineers. It's to come up with example for the divergence theorem in fluid dynamics, but finding a very ...
shuhalo's user avatar
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3 votes
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Help for evaluating the line integrals with Green's Theorem

Earlier, when I scrolled the Instagram posts I found a mathematical problem uploaded by The Vegan Math Guy like the following because this problem looks interesting to me to be solved. The ...
Arthur Kangdani's user avatar
1 vote
0 answers
75 views

Help using Green's theorem to find volume of function inside a polygon

For my work, I am trying to find the volume under a two dimensional function $f(x,y)$ bounded by a polygon of $n$ vertices. My dim memory of undergrad is that Green's theorem is the way to go for this,...
Ingolifs's user avatar
  • 381
2 votes
1 answer
300 views

How to calculate the area of the region enclosed by $x^3+y^3=3axy$? [duplicate]

Let $y=tx,$ after calculation, I get $$x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3}.$$ Use the Green Formula, the area is equal to $$\frac{1}{2}\oint_{\Gamma}xdy-ydx.$$ My question is: how to determine ...
Ychen's user avatar
  • 577
1 vote
2 answers
119 views

Can i use Green's theorem to calculate the area of an abstract triangle on a plane?

I want to see some examples of Green's theorem used to calculate the area of some simple 2D shapes, but i haven't encountered a lot of them. My goal is to find (or study) a general procedure for ...
Simón Flavio Ibañez's user avatar
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0 answers
43 views

Using Green's Theorem in a region with the origin

I have to evaluate the line integral in the field $$F(x,y)=(-2y+\sqrt{4-x^2},\ln(y)-x)$$ over the circle $x^2+y^2=5^2$. Since the origin is inside this circle, I can't use Green's Theorem. I'd like to ...
mvfs314's user avatar
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11 votes
1 answer
164 views

How to evaluate $\displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx$ where $f(x)=\cos(x)+\sin(f(x))$

So I want to find the area of this circular looking thing. I had the following thought process in solving it. Consider the implicit derivative of the function. $\begin{align} y &= \cos(x)+\sin(y) ...
Nεo Pλατo's user avatar
1 vote
0 answers
200 views

Use Green's Theorem to prove Green's first identity

The exercise asks Use Green's Theorem in the form of equation 13 to prove Green's First Identity: $$ \iint\limits_D f \nabla^2 g \, dA = \oint_C f(\nabla g) \cdot \mathbf{n} \, ds - \iint\limits_D \...
iwjueph94rgytbhr's user avatar
2 votes
1 answer
155 views

Stokes' Theorem and perimeter

As a consequence of Stokes' Theorem it seems that the perimeter of a closed curve $C$ can be obtained by choosing $F$ to be the vector field formed by rewriting the unit tangent $T$ of $C$ as a ...
Simon M's user avatar
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Vector field is conservative: proof verification

I've been trying to prove that this vector field: $$ \vec{F}=\left(\frac{y}{\left(x-1\right)^{2}+y^{2}},\frac{1-x}{\left(x-1\right)^{2}+y^{2}}\right) $$ Is conservative in: $$ D=\left\{ \left(x,y\...
AnonA's user avatar
  • 65
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0 answers
37 views

Does Green's theorem need conditions before usage?

I want to evaluate $\frac{1}{2\pi}\oint_C\dfrac{-y\,dx+x\,dy}{x^2+y^2}$ clockwise around the square with vertices $(-1,-1)$, $(-1,1)$, $(1,1)$ and $(1,-1).$ Obviously $C$ is closed. So if we use Green'...
Mary Jane's user avatar
0 votes
1 answer
119 views

Solution of Neumann problem for Laplace equation.

I have the following problem: Let $ u $ be in $ C^2(\Omega) $ and in $ C^1(\overline{\Omega}) $, where $ \Omega $ is a normal bounded domain in $ R^n $, and suppose that \begin{equation*} \begin{split}...
Berban's user avatar
  • 29
0 votes
1 answer
89 views

Seeking help in understanding the proof of the mean value property for harmonic functions

I am currently trying to understand the proof of the mean value property from 'Harmonic Function Theory' by Axler, Bourdon, and Ramet. Mean-Value Property: If $u$ is harmonic on $\bar{B}(a, r)$, then $...
RiXaTorAgu's user avatar
3 votes
1 answer
113 views

Green's Theorem does not check out, spot the mistake.

We have a square on the plane of sides 2 from (-1,-1) to (1,1), and $P(x,y)=x^2+y^2,Q(x,y)=2x^2y$. $$ \oint_L (x^2+y^2)dx+2x^2ydy=2\int_{-1}^1(x^2+1)dx+2\int_{-1}^12ydy=\frac{16}{3}+0=\frac{16}{3} $$ ...
user34909's user avatar
  • 143
0 votes
2 answers
99 views

Applying Green's theorem to this integral

Let the area outside the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ bounded by the curves $\frac{x^2}{25}+\frac{y^2}{16}=1(y\leq 0)$,$x+y=5$ and $y-x=5$ be $R$, $C$ is the outer bound of $R$ and $C_1$ be ...
user avatar
2 votes
1 answer
94 views

How do we use the right hand rule for Stokes' theorem?

Let $C$ be the intersection curve between the plane $z = 10 - x - y$ and the cylinder $x^2+y^2 = 1$, oriented such that the projection of the curve onto the xy-plane is positively oriented. Determine ...
Need_MathHelp's user avatar
0 votes
1 answer
83 views

Determine the integral using laplacian equation and Green's theorem

Consider a simple closed curve $C$ in the plane (positively oriented) with $p$ and $q$ as two points inside $C$. Let $f$ be a given function that is differentiable in $\mathbb{R}^2 \setminus \{p, q\}$ ...
Need_MathHelp's user avatar
3 votes
0 answers
166 views

Is there a connection between shoelace formula and Stokes theorem?

The shoelace-formula is a method to calculate the area of a polygon. It is given as $$ A = 1/2 \sum_i{(x_i-x_{i+1})*(y_i+y_{i+1})} $$ for cyclical $i$. Expanding the product yields the terms $x_i y_i -...
GammaSQ's user avatar
  • 181
0 votes
0 answers
60 views

Using Green's theorem to calculate a line integral by first closing the region then removing the added part

Calculate $$\int_\gamma (2e^{{(y-2x)}^2}-y)dx+(-e^{({y-2x})^2}+2x)dy$$ where $\gamma$ is the section of the curve $y=x^2$ from the point $(0,0)$ to $(2,4)$. A tip I got was to first close the region ...
ShootinLemons's user avatar
0 votes
1 answer
195 views

How to use Green's theorem on unit circle

Compute $∫_{\gamma} −y^3 dx + x^3 dy$, where $\gamma$ is the positively oriented edge of the unit circle. So I used Green's theorem and got $$\begin{align}∫_{\gamma} −y^3 dx + x^3 dy &=\iint_{D} ...
Need_MathHelp's user avatar
1 vote
1 answer
76 views

Green's Theorem - Applying a translation to the vector field to make the integral easier to evaluate

I have to evaluate a line integral using Green's theorem: $$\overrightarrow{F}(x,y)=\langle y-\cos{y}, x\sin{y} \rangle$$ The curve $C$ is the circle $(x-3)^2+(y+4)^2=4$, oriented clockwise. It would ...
Leonardo Cucinotta's user avatar
1 vote
0 answers
40 views

Green's Theorem circle in a circle (hole) when both are traversed in the same direction

Im struggling to understand how to apply Green's theorem in the case where you have a hole in a region which is traversed in the same direction as the exterior. For a workable example I want to ...
zrn's user avatar
  • 53
0 votes
0 answers
24 views

inferring sign of a double integral over a general region D from double integrals of all boxes that lie inside D and containing D

Suppose a double integral $$ I_{D}:=\iint\limits_D f(x,y) dx dy $$ over a specific type of region $D=\{(x,y) : x_{\min}\leq x \leq x_{\max}, h_1(x) \leq y \leq h_2(x)\}$. Note that $h_1, h_2$ are ...
GSecer's user avatar
  • 71
2 votes
2 answers
82 views

Calculating line integral on a vector field, help me find the mistake

Alright, so, I have vector field: $F=[p(x,y), q(x,y)]=[y^3+e^{x^2}, x^3+{\tan}^2y]$. I need to calculate $\oint_Lpdx+qdy$, where $L: x^2+y^2+4y=0$. I transform it to $x^2 + (y+2)^2 = 4$, i.e. a circle ...
bsqpt's user avatar
  • 23
0 votes
0 answers
48 views

Find $ \oint_c \frac{-y}{x^2+4y^2}dx+\frac{x}{x^2+4y^2}dy$ when $c$ is the unit circle.

Find $$ \oint_c \frac{-y}{x^2+4y^2}dx+\frac{x}{x^2+4y^2}dy$$ when $c$ is the unit circle (Counterclockwise). My attempt: Denote $P=\frac{-y}{x^2+4y^2}, Q=\frac{x}{x^2+4y^2}$. $Q_x=\frac{4y^2-x^2}{(x^...
Algo's user avatar
  • 2,322
1 vote
1 answer
58 views

Weak formulation of a stationary Schrodinger equation on $ H^1_W(\mathbb{R}^n) $

I need to find the weak formulation of this equation on $ H^1_W(\mathbb{R}^n) $ the weighted Sobolev space. $$ \left(\dfrac{-1}{2m}\Delta + V(x) - \lambda\right)u = f $$ With $V(x)$ bounded from $\...
tareqath's user avatar
  • 107
0 votes
0 answers
162 views

How to integrate $(x+y) \ dx + (x-y) \ dy$ over an ellipse?

Show that the ellipse defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ in $\mathbb{R}^2$ is a submanifold and orient it counterclockwise. Compute $\int_M \omega$ for $\omega := (x+y) \ ...
3nondatur's user avatar
  • 4,178
0 votes
1 answer
40 views

Solve $\int_{A} \frac{x^2y + y^3 - y}{x^2 + y^2} \, dx + \frac{x^3 + xy^2 + x}{x^2 + y^2} \, dy$

Solve $\displaystyle \int_{A} \frac{x^2y + y^3 - y}{x^2 + y^2} \, dx + \frac{x^3 + xy^2 + x}{x^2 + y^2} \, dy $ , $A$ is the unit circle. My attempt: $\displaystyle \int_{A} \frac{x^2y + ...
Algo's user avatar
  • 2,322
1 vote
1 answer
34 views

Green's theorem with vector field

I am given the vector field $$\vec{F} = (9x^2y+3y^3+2e^x,6e^{y^2}+225x)$$ and $C_a$ a circle of radius a and the center of origin (0,0), counter clockwise. I am trying to calculate $$\oint_{C_1} \vec{...
SpaceNugget's user avatar
-3 votes
1 answer
71 views

Question related to green's theorem

I am a not experienced in linear algebra, and I am not really sure how to tackle this problem. Thanks in advanced. Show that, $$\nabla u \cdot \mathbf m=(\mathbf m \cdot \mathbf n)\nabla u \cdot\...
Maram's user avatar
  • 11
0 votes
1 answer
47 views

Computing the line integral using Green's formula

$$\int_{L}{{e^{-(x^2+y^2)}}(\cos2xydx+\sin2xydy)}$$, where L is this circle $x^2+y^2 = R^2$. Since I have to use Green's formula, I computed $\frac{\partial{P}}{\partial{y}}$ and $\frac{\partial{Q}}{\...
upper's user avatar
  • 107
0 votes
2 answers
79 views

Question on Green's Theorem

For $F=yx^3\hat{i}+y^2\hat{j}$ compute the line integral $\int_C F\cdot dr$ where $C$ is the curve $y = x^2$ for $0\le x\le 1$. Can this be solved with Green's Theorem? I am aware of the method where ...
user112167's user avatar
1 vote
1 answer
49 views

Where did I do wrong in solution of $\int_C\left(x \mathrm{e}^{y^2}-2 y\right) \mathrm{d} x+\left(x^2-1\right) y \mathrm{e}^{y^2} \mathrm{~d} y$

I had a wrong answer when solving $$ \int_C\left(x \mathrm{e}^{y^2}-2 y\right) \mathrm{d} x+\left(x^2-1\right) y \mathrm{e}^{y^2} \mathrm{~d} y $$ where $$ C:y=\sqrt{2x-x^2} $$ My first thought was, ...
mingzi xingshi's user avatar
3 votes
1 answer
293 views

Calculating a line integral using Green's theorem in a region with a singularity

Problem: Calculate the line integral $$ \int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1} $$ where $A$ is the line $|x|+|y|=4$, travelling clockwise and making one rotation. Answer: $-2\pi$ Solution: The ...
Tsiolkovsky's user avatar

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