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 $C$.
3
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2answers
52 views
Find a simple and smooth curve $C$ such that $\displaystyle \int_C\vec{F}\cdot d\vec{r}$ gets its maximum value
I've been trying to solve this problem for a while, but for too long couldn't I continue my partial solution. I would be glad if you could shed some light on my solution.
The task:
Given the vector ...
1
vote
1answer
73 views
Find the curve of the maximum value of work done?
Suppose $C$ is a simple close curve (i.e. it doesn’t intersect itself) in the first quadrant. If $F = (y^2/2 + x^2y, -x^2 + 8x)$, find the curve that produces the maximum amount of work done by $F$.
...
1
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0answers
18 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
$$(\nabla_{\Sigma}\cdot w ,v)_{\Sigma}=(n_{\Gamma}\cdot ...
0
votes
0answers
15 views
Difference between (Variation of Parameters) and Green Function.
I'm confused with both of them. Variation of parameter and green function. I read on some papers that using green function on pde or ode is just like variation of parameter.
But wait, in some videos, ...
0
votes
1answer
22 views
Using Green's theorem to compute integral on curve
Prove:
$$\ \int_C (\sin x - y^2)dx +(x-y \tan^{-1}(y^2))dy = 2.4 $$ where $\ C $ is the curve from $\ (1,2) $ to $\ (-1,2) $ on $\ y = x^2 + 1 $
Using green's theorem
$$\ \int \int_D (Q_x - P_y)dx ...
0
votes
2answers
36 views
Calculate line integral L: $y=sinx$, $y=0$, $0\le x \le \pi$
There is an example to calculate the line integral $\oint_{L}P(x,y)dx+Q(x,y)dy$
The contour $L$: $y=\sin x$, $y=0$, $0\le x \le \pi$
$P(x,y)=e^{x}y$, $Q(x,y)=e^{x}$
The calculation has to be ...
0
votes
1answer
26 views
Green's formula for the Laplacian defined in a neighborhood of the surface
Source: https://arxiv.org/pdf/1705.00069.pdf
On page 4, it says that the surface Laplacian of a function $u$ (I will use different letters here) defined on a neighborhood of the boundary $\partial M$...
0
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0answers
15 views
Evaluate the line integral using two methods : directly and green
Consider the line integral
question
bounds
i get how to find the parametize equations.
what i dont get is how they find the upper and lower bounds for the integrals.
so for c1= y=0 ; x =t ; 0
c3 ...
1
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1answer
20 views
Closed loops on nonconservative vector field
Consider the following (very simple) nonconservative vector field:
$$V = (y, -x)$$
sketched in the following figure:
Roughly speaking, the integral along the central red circle is obviously $\neq 0$...
0
votes
1answer
70 views
What is the result of the integration by parts of $\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega$?
$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$
where $\Omega \subset \mathbb{R}^2$ is a bounded domain with Lipschitz continuous and piecewise smooth boundary $\Gamma:=\partial \Omega$, $...
0
votes
1answer
45 views
How to evaluate integral of (x - y)(dx + dy) with Green's Theorem?
I want to evaluate the integral $\int(x - y)(dx + dy)$ along curve C where C is the semicircular part of $x^2 + y^2 = 4$ above $y = x$ from $(-\sqrt2, -\sqrt2)$ to $(\sqrt2, \sqrt2)$ using Green's ...
2
votes
1answer
44 views
Evaluating $I = \iint_D (x+y)\, dy\,dx$ using Green's Theorem
Let $D$ be the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$.
I want to evaluate the following integral
$$I = \iint_D (x+y)\, dy\,dx$$
using two methods: by direct integration, and by ...
2
votes
1answer
32 views
Green's theorem over an annulus
I need help with this problem:
Verify Green's Theorem in the plane where $S$ is the annulus $\{(x,y)\in\mathbb{R^2}|a^2\leq x^2+y^2\leq b^2\}$ and
$F(x,y)=\left(\frac{-y}{\sqrt{x^2+y^2}},\...
1
vote
1answer
50 views
Green theorem intuition
What I have a hard time understanding is the connection between line integrals of vector fields and Greens theorem.
It was explained that taking lines integrals of parametrized curves is to be ...
3
votes
1answer
48 views
Is it “Valid” to prove Stokes' Theorem with Green's Theorem?
In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in ...
1
vote
1answer
73 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
$$ \int_\gamma y^3 \sin^2(x) \, dx - x^5 \cos^2(y) \,...
0
votes
1answer
20 views
What is a region of type 3 with regards to Green's Theorem?
I understand that a region of type 1 is where two curves are connected by two vertical lines and that a region of type 2 is where two curves are connected by two horizontal lines. But what is a region ...
0
votes
1answer
32 views
Path Integral equals zero on non conservative field
I was doing some excercises and I was asked to compute the line integral along certain path. I used greens formula to calculate the work. When computing the integral I had to divide the domain in two ...
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0answers
11 views
Greens function representation of nonlinear Poisson equation
Let $L$ be an operator and suppose the Green's function exists. That is there exist a function $G$ such that $LG=\delta$ where $\delta$ is the Dirac delta function.
If $L$ is linear, one can represent ...
1
vote
2answers
50 views
Green's Theorem confusion
I have two equivalent forms of Green's theorem, namely
$$
\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}+\...
0
votes
0answers
18 views
Green's theorem Area formula additional cases
In this thread, the OP states 3 area formulas and asks for proof. The answer given uses 3 specific cases where ∂Q/∂x − ∂P/∂y=1. I see why these 3 cases give the area formulas, but what about other ...
1
vote
1answer
19 views
For Green's theorem, why is the region of integration of the line integral a weird partial derivative character?
Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem?
$$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$
...
0
votes
1answer
16 views
What does it mean for a region to be simultaneously a region of type 1 and type 2?
I am going through a proof of Green's Theorem for a simple region and I understand the mathematics taking place but do not understand the origins.
'Regions that are simultaneously of type I and II ...
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0answers
15 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. ...
0
votes
0answers
36 views
Integral on surface and boundary
Let $\Omega$ be connected bounded open set in $\mathbb{R}^{n}$. Let $U:\Omega\rightarrow \mathbb{R}^{n}$ be a $C^{1}$ vector field. The divergence theorem is given
\begin{align}
\int_{\Omega} \nabla\...
0
votes
0answers
14 views
Can someone explain to me the jump between steps in the bottom two lines of this proof (not yet totally complete)?
Suppose we have a region $G$ that is bounded by the straight lines $x=a$, $x=b$, $y=c$ and by an arc $y = f(x)$ (which lies above $y=c$) where $a \leq x \leq b$.
If $f$, $P(x,y)$ and $Q(x,y)$ are all ...
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0answers
33 views
Show that $\int_{\Omega}\det(DF(x))dx = \det(M)\mathrm{area}(\Omega)$.
Let $\Omega$ be a bounded open in $\mathbb{R}^{2}$ such that $\partial \Omega$ is a $C^{1}$ curve, $F: \mathbb{R}^{2} \to \mathbb{R}^{2}$ a twice differentiable function. For $x = (x_{1},x_{2}) \in \...
0
votes
1answer
71 views
find the simple closed curve of $F(x,y) = (y^3-6y)i + (6x-x^3)j$ using Green's Theorem which will have the largest positive value
$F(x,y) = (y^3-6y)i + (6x-x^3)j$
a. Using Green's Theorem, find the simple closed curve C for which the integral
$ ∳F \cdot dr $ (with positive orientation) will have the largest positive value.
b....
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0answers
49 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 ...
0
votes
2answers
47 views
Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$
Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$
my question is , after calculating the integral using green theorem ...
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0answers
37 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
3answers
41 views
how to calculate $\int_{C}(2x^2+1)e^{x^2+y^2}dx+(2xy)e^{x^2+y^2}dy$ using Green theorem
Compute $\int_{C}(2x^2+1)e^{x^2+y^2}dx+(2xy)e^{x^2+y^2}dy$ where $C$ connects $(1,0)$ to $(0,1)$ by a straight line segment.
I tried to use green theorem since $Q_x = P_y$ so $\int \vec F_\dot{}\vec ...
0
votes
2answers
62 views
how to calculate $\int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2}$ using green theorm or directly
Calculate $$\int_C \frac{2xy^2dx-2yx^2dy}{x^2+y^2},$$ where $C$ is the ellipse $3x^2 +5y^2 = 1$ taken in the positive direction.
I tried to calculate the integral using green theorm.
now i need to ...
1
vote
1answer
35 views
Integral of a differential form along a non-defined path
Let $R>0$ and $\Omega=\{(x,y)\in \mathbb{R^2}:x^2+y^2<R^2,y>0\}$. Consider also $\omega (x,y)=x^2dx+2xydy$. My goal is to prove that $\int_{\partial_{+}\Omega}\omega=\frac{4}{3}R^3$, where $\...
0
votes
1answer
37 views
Line integral in proof of Green's theorem
In wikipedia page about Green's theorem the following equality appears:
$$
\int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx
$$
I do not understand it. Wikipedia page about line integral defines ...
1
vote
1answer
68 views
Show that $F_x(x,y) = P(x,y), \ F_y(x,y) = Q(x,y)$.
In Complex Variables and Applications by Brown and Churchill it comes:
When the point $(x_0,y_0)$ is kept fixed and $(x , y)$ is allowed to vary throughout a simply connected domain $D$, the ...
0
votes
0answers
18 views
Green’s theorem double into line integral
How I can convert this double integral into line $\int \int \frac{dx dy}{y^2} $ we have $\frac{dQ}{dx}-\frac{dP}{dy}=\frac{1}{y^2}$ how to find $Q, P$ functions? Any two functions works? $P=1, Q=\frac{...
1
vote
1answer
29 views
Trouble calculating line integral using Green's theorem, complicated integral.
"Calculate line integral of scalar function [$y(e^x) -1]dx + [e^x]dy$ over curve $C$, where $C$ is the semicircle through $(0, 10), (10, 0)$, and $(0, 10)$"
I plan on using Green's theorem, and since ...
1
vote
1answer
119 views
Using Green's Theorem to find the flux
$F(x,y)=\langle y^2+e^x,x^2+e^y\rangle$. Using green's theorem in its circulation and flux forms, determine the flux and circulation of $F$ around the triangle $T$, where $T$ is the triangle with ...
0
votes
2answers
105 views
Verifying Green's theorem for a function
Let
$G = \{ (x,y) \in \mathbb{R}^2 : x^2+4y^2 >1, x^2+y^2 < 4 \} $
$ \int_G x^2+y^2 d(x,y) $
I want to verify Green's Theorem :
$ \oint_{ \partial G } f n ds = \int_G \operatorname{div}f\, ...
-1
votes
1answer
16 views
green theorem statement's meaning
Vector Calculus sector of 6.2
17. D is always on the left as we travel along C (C is the path of D)
What is the meaning of the above statement? I can't understand ;(
0
votes
1answer
155 views
Area of a simple closed curve
Let C be a simple closed curve in a region where Green's Theorem holds. Show that the area of the region is:
\begin{equation}
A=\int_{C}xdy=-\int_{C}ydx
\end{equation}
Green's theorem for area ...
0
votes
0answers
31 views
Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field
Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field
I have thought to do the following to solve this:
...
0
votes
1answer
54 views
Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ along a path
Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ when a particle moves from the origin, along the $x$ axis to the $(1,0)$, then on the line segment that joins the $(1,0)$ ...
0
votes
2answers
22 views
Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined
Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined by the inequalities $-a\leq x\leq a$, $-a\leq y\leq a$, ...
1
vote
1answer
43 views
Why is $ I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\zeta) d\zeta = -2i \int_{\partial \Omega} \frac{1}{z-\zeta}\partial u(\zeta) d\zeta. $?
I am unsure of some notation and also how a particular identity was derived. I read in a paper that because $\Delta = 4 \partial \bar \partial$ we have
$$
I = \int_\Omega \frac{1}{z-\zeta}\Delta u(\...
0
votes
2answers
69 views
Line Integral in second quadrant of Unit Circle
If I am asked to compute
$$\int_c F . dr$$
Where
$$F(x,y) = <d/dx f(x, y), d/dy f(x,y)>$$
and
$$f(x,y) =\sin(x^3 + y^3)$$
and C is the portion of the unit circle in the second quadrant, ...
0
votes
1answer
24 views
Variational Formulation - inhomogeneous
I'm not sure how to get started with the following. Consider,
$- \Delta u=f$ in $\Omega$
$u=u_o$ on $\Gamma$
I need to find a $u \in V(u_o)$ such that
$a(u,v)=(f,v)$ $\forall v\in H^1_o$ where
$...
1
vote
1answer
66 views
Green's Formula in the case where $M = x^2 - y^2$ and $N=2xy$
Question: Test Green's Formula in the case where $M = x^2 - y^2$ and $N=2xy$ and $\Omega$ is the triangle with vertices $(0, 0), (1,1), (2, 0)$.
My attempt: $$\iint_\Omega \Big(\dfrac{\partial M}{\...
0
votes
1answer
141 views
Evaluate using Green's Theorem over ellipse
I have the answer to a problem and am trying to understand the steps to get to that answer. The problem is $\oint_C(x+2y)dx+(y-2x)dy$ around the ellipse C, defined by $x=4cos\theta, y=3sin\theta, 0\...
