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 $C$.
570
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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: $(0\le x \le 1, 0), (1,0 \le y \le 1), (1 \le x \le 0, 1),(0,1\le y \le 0)$
...
0
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1
answer
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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 ...
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Finding area with Green's theorem [duplicate]
Given an area, be it an astroid or a fish-like shape, how is Green's theorem helpful to these situations, if there is no given vector field?
We were given the following formulas, said to have been ...
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0
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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 ...
1
vote
1
answer
32
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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: ...
1
vote
2
answers
127
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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.
...
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0
answers
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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+...
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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:
$$ \iiint_O {\nabla \cdot {\bf F}}\ dV = \iint_{S=\partial O} ({\bf F}\cdot \hat{{\bf n}...
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1
answer
48
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Bounded second derivative also bounds the function
I've been struggling with this problem for a few days:
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as
$f(x,y)=xg(y)-yg(x)$, where $g:\mathbb{R}\rightarrow \mathbb{R}$ is
such that $g\in C^2$, $...
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1
answer
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Is the Normal vector is pointing inside or outside?
Good morning,
I am posting this again because I am having trouble to understand this post (How do I check if the normal vector is pointing inside or outside?).
Here is my question :
I have the ...
1
vote
0
answers
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How to transform area integral $\int_{D} \omega^2 \ dx \ dy$ into boundary integral $\oint_{C} \square \ ds$?
Let $\omega$ be a function that satisfies the Laplace's equation
$$\nabla^2 \omega = 0$$
The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the ...
0
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0
answers
40
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Stokes / Green's theorem with non-regular regions
One statement of Green's Theorem (Stewart) I have seen is:
Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$. If $\mathbf{F}...
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Line integral exercise of a real vector field
I'm trying to solve the following exercise of a vector field over line integral:
$$\int\limits_C\frac{-y}{4x^2+9y^2}dx+\frac{x}{4x^2+9y^2}dy,$$
where $C$ is the closed curve formed by the equations $y=...
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Weaker version of Goursats theroem
I have a homework problem that states:
Let $\Omega \subseteq \mathbb{C}$ open. Suppose that $f:\Omega \rightarrow \mathbb{C}$ is holomorphic and $C^1$. Show that:
$\int_{T}f dz = 0$
Where $T$ is an ...
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0
answers
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Green's theorem example application for engineers and physicists.
I am looking for example applications of Green's theorem (in $2D$) that appeal to physicists or engineers. It's to come up with example for the divergence theorem in fluid dynamics, but finding a very ...
3
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Help for evaluating the line integrals with Green's Theorem
Earlier, when I scrolled the Instagram posts I found a mathematical problem uploaded by The Vegan Math Guy like the following because this problem looks interesting to me to be solved. The ...
1
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0
answers
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Help using Green's theorem to find volume of function inside a polygon
For my work, I am trying to find the volume under a two dimensional function $f(x,y)$ bounded by a polygon of $n$ vertices.
My dim memory of undergrad is that Green's theorem is the way to go for this,...
2
votes
1
answer
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How to calculate the area of the region enclosed by $x^3+y^3=3axy$? [duplicate]
Let $y=tx,$ after calculation, I get
$$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.$$
My question is: how to determine ...
1
vote
2
answers
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Can i use Green's theorem to calculate the area of an abstract triangle on a plane?
I want to see some examples of Green's theorem used to calculate the area of some simple 2D shapes, but i haven't encountered a lot of them. My goal is to find (or study) a general procedure for ...
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answers
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Using Green's Theorem in a region with the origin
I have to evaluate the line integral in the field
$$F(x,y)=(-2y+\sqrt{4-x^2},\ln(y)-x)$$
over the circle $x^2+y^2=5^2$. Since the origin is inside this circle, I can't use Green's Theorem.
I'd like to ...
11
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1
answer
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How to evaluate $\displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx$ where $f(x)=\cos(x)+\sin(f(x))$
So I want to find the area of this circular looking thing.
I had the following thought process in solving it.
Consider the implicit derivative of the function.
$\begin{align} y &= \cos(x)+\sin(y) ...
1
vote
0
answers
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Use Green's Theorem to prove Green's first identity
The exercise asks
Use Green's Theorem in the form of equation 13 to prove Green's First Identity:
$$
\iint\limits_D f \nabla^2 g \, dA = \oint_C f(\nabla g) \cdot \mathbf{n} \, ds - \iint\limits_D \...
2
votes
1
answer
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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 ...
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0
answers
60
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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:
$$
D=\left\{ \left(x,y\...
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0
answers
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Does Green's theorem need conditions before usage?
I want to evaluate $\frac{1}{2\pi}\oint_C\dfrac{-y\,dx+x\,dy}{x^2+y^2}$ clockwise around the square with vertices $(-1,-1)$, $(-1,1)$, $(1,1)$ and $(1,-1).$
Obviously $C$ is closed. So if we use Green'...
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1
answer
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Solution of Neumann problem for Laplace equation.
I have the following problem:
Let $ u $ be in $ C^2(\Omega) $ and in $ C^1(\overline{\Omega}) $, where $ \Omega $ is a normal bounded domain in $ R^n $, and suppose that
\begin{equation*}
\begin{split}...
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1
answer
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Seeking help in understanding the proof of the mean value property for harmonic functions
I am currently trying to understand the proof of the mean value property from 'Harmonic Function Theory' by Axler, Bourdon, and Ramet.
Mean-Value Property: If $u$ is harmonic on $\bar{B}(a, r)$, then $...
3
votes
1
answer
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Green's Theorem does not check out, spot the mistake.
We have a square on the plane of sides 2 from (-1,-1) to (1,1), and $P(x,y)=x^2+y^2,Q(x,y)=2x^2y$.
$$
\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}
$$
...
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votes
2
answers
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Applying Green's theorem to this integral
Let the area outside the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ bounded by the curves $\frac{x^2}{25}+\frac{y^2}{16}=1(y\leq 0)$,$x+y=5$ and $y-x=5$ be $R$, $C$ is the outer bound of $R$ and $C_1$ be ...
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1
answer
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How do we use the right hand rule for Stokes' theorem?
Let $C$ be the intersection curve between the plane $z = 10 - x - y$ and the cylinder $x^2+y^2 = 1$, oriented such that the projection of the curve onto the xy-plane is positively oriented. Determine ...
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1
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Determine the integral using laplacian equation and Green's theorem
Consider a simple closed curve $C$ in the plane (positively oriented) with $p$ and $q$ as two points inside $C$. Let $f$ be a given function that is differentiable in $\mathbb{R}^2 \setminus \{p, q\}$ ...
3
votes
0
answers
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Is there a connection between shoelace formula and Stokes theorem?
The shoelace-formula is a method to calculate the area of a polygon. It is given as
$$
A = 1/2 \sum_i{(x_i-x_{i+1})*(y_i+y_{i+1})}
$$
for cyclical $i$. Expanding the product yields the terms $x_i y_i -...
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0
answers
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Using Green's theorem to calculate a line integral by first closing the region then removing the added part
Calculate
$$\int_\gamma (2e^{{(y-2x)}^2}-y)dx+(-e^{({y-2x})^2}+2x)dy$$
where $\gamma$ is the section of the curve $y=x^2$ from the point $(0,0)$ to $(2,4)$.
A tip I got was to first close the region ...
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votes
1
answer
172
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How to use Green's theorem on unit circle
Compute $∫_{\gamma} −y^3 dx + x^3 dy$, where $\gamma$ is the positively oriented edge of the unit circle.
So I used Green's theorem and got
$$\begin{align}∫_{\gamma} −y^3 dx + x^3 dy
&=\iint_{D} ...
1
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1
answer
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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 ...
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0
answers
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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 ...
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0
answers
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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 ...
2
votes
2
answers
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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 ...
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0
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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^...
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1
answer
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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 $\...
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0
answers
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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) \ ...
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votes
1
answer
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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 + ...
1
vote
1
answer
34
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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 $$\oint_{C_1} \vec{...
-3
votes
1
answer
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Question related to green's theorem
I am a not experienced in linear algebra, and I am not really sure how to tackle this problem. Thanks in advanced.
Show that,
$$\nabla u \cdot \mathbf m=(\mathbf m \cdot \mathbf n)\nabla u \cdot\...
0
votes
1
answer
47
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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}}{\...
0
votes
2
answers
76
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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 ...
1
vote
1
answer
49
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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, ...
3
votes
1
answer
285
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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 ...
0
votes
0
answers
31
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Does the starting/end point matter in line intergral of vector fields over a closed curve
Suppose I have a vector field $F = (F_1, F_2):\mathbb{R}^2 \to \mathbb{R}^2$ and $C$ a positively oriented smooth simple closed curve. Then I know the definition of
$$
\int_C F_1 dx + F_2 dy = \int_{...
6
votes
2
answers
471
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How does Green's theorem and Stokes' theorem generalize the fundamental theorem of Calculus
I've read in few places that Green's theorem
$$
\oint_C L dx + M dy = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx dy
$$
is a generalization of fundamental ...