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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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1answer
38 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 ...
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1answer
69 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 ...
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20 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{...
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1answer
30 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 ...
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1answer
141 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 ...
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2answers
116 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\, ...
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1answer
17 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 ;(
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1answer
254 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 ...
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0answers
40 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: ...
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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 $(1,0)$ ...
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2answers
23 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 $-a\leq x\leq a$, $-a\leq y\leq a$, ...
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1answer
45 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(\...
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2answers
77 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, ...
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1answer
25 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 $...
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1answer
71 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}{\...
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1answer
166 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\...
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2answers
44 views

Which theorem says that $\displaystyle \int_{f_1(x)}^{f_2(x)}\frac{\partial M}{\partial y}\,dy=M(x,f(x_2))-M(x,f(x_1))$?

I want to explain where this equality comes from. I'm working with a proof of Green's theorem. Thanks very much.
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207 views

Green's Theorem on a region with holes

I'm trying to understand Green's Theorem and its applications and there's something that just doesn't make sense to me. Consider the following shape and parametrization: Why can't I simply use a ...
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76 views

Anti-holomorphic derivative on 1/z

We know the fact that $\partial_{\bar z} (1/z) \propto \delta(z, \bar z)$. But I am slightly confused about the proof. I have seen a few of them, and of course they use the Green's theorem, or the ...
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2answers
435 views

Verify Stokes' Theorem for $v=zi+xj+yk$ over the hemispherical surface $x^2+y^2+z^2=1$ and $z \gt 0$.

Verify Stokes theorem if $v=zi+xj+yk$ (where $i,j,k$ are the identity vectors for the $x,y,z$ axis) is taken over the hemispherical surface $x^2+y^2+z^2=1$ and $z \gt 0$. Stokes theorem being: $$\int\...
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1answer
62 views

Application of Green's theorem in a plane

Apply Green's theorem in a plane to evaluate $\displaystyle\int(2x^2-y^2)dx + (x^2+y^2)dy$, where $C$ is the curve enclosed by the semi-circle $x^2+y^2=1$ and the $x$-axis. I've done this already $$ ...
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1answer
17 views

Show that $F$ has a maximum in $\mathcal{C}$. $\mathcal{C}$ set of curves.

Let $\mathcal{C}$ the set of all regular, simple and closed curves in $\mathbb{R}^2$. Consider the function $F: \mathcal{C} \to \mathbb{R}$ defined by $$F(\gamma) := \int_\gamma(y^3 - y + \sin(x^5)) ...
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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{\...
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1answer
124 views

Limacon curve and its osculating circle

Consider the Limacon: $\gamma(t)=((1+3cost)cost, (1+3cost)sint)$. (i) Compute $A(\gamma)=\frac{1}{2}\int_\gamma (x\frac{dy}{dt}-y\frac{dx}{dt})dt$. (ii) Determine the osculating circle $C$ at $(4,0)$...
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1answer
197 views

use Green theorem to evaluate line integral:

In the problem below, parametrize the plane curves below in such a way that it traversed only once , its unit normal vector towards the interior of the bounded region it enclose. Then use Green ...
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1answer
40 views

Evaluate $\int_{\Gamma}xy^2dx+xydy$ on $\Gamma={y=x^2}$

Evaluate $\int_{\Gamma}xy^2dx+xydy$ on $\Gamma=\{(x,y)\in\mathbb{R}^2:y=x^2,x\in[-1,1]\}$ with orientation clockwise using Green theorem So $\Gamma$ is a parabola to use Green we have to close the ...
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3answers
81 views

Evaluating area of $\frac{3}{4}$ of a disc

Evaluate $$\int_{\gamma_1\cup \gamma_2}xdx+x^2ydy$$ Where $\gamma_1(t)=(2\cos t,2\sin t),t\in [-\frac{\pi}{2},\pi]$ $\gamma_2(t)=(\cos t,\sin t),t\in [-\frac{\pi}{2},\pi]$ Using ...
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1answer
62 views

Integral of differential forms $\int (-y+\sin x^2)dx + xdy$.

I should calculate $$\int (-y+\sin x^2)dx + xdy$$ on the curve $c=c_{1}+c_{2}-c_{3}-c_{4}$ where it doesn't give me any parametrisation mappings; only the normal $$ \begin{cases} c_{1}:& [0,...
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1answer
118 views

How to deduce Green's formulas from Gauss-Green's theorem?

Gauss-Green Theorem Gauss-Green Green's Formulas Green What I'm trying to do is to demonstrate that all Green's formulas follow from the Gauss-Green theorem, which are all given above. I am aware ...
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1answer
33 views

Some questions regarding a formula that's supposed to be the integral formula of Gauss and Green.

According to my professor, this formula is called the integral formula of Gauss and Green. I've tried searching for it online but I could find nothing similar to it. B here is a region that's bounded ...
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1answer
286 views

Using Green's Theorem to Calculate the Counter-Clockwise Circulation for the Field $\mathbf{F}$ and Curve $C$.

I have this problem Use Green’s Theorem to find the counter-clockwise circulation for the field $\mathbf{F}$ and curve $C$. with this image Green's Theorem says that the counter-clockwise ...
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1answer
34 views

$\int_V \left( g\nabla^2f-f \nabla^2g \right) dV=\int_S \left( g\nabla f-f \nabla g \right) \cdot u_n \, dS$ not depend on time

I' m wondering why the following relationship, known as Green's identity, doesn't depends on time. Let $f(x,y,z,t)$ and $g=\frac{e^{ikr}}{r}$ so $$\int_V \left( g\nabla^2f-f \nabla^2g \right) dV=\...
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0answers
83 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{\...
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2answers
36 views

Exercises of aplication of Green's theorem

Use Green's Theorem to find the limited area above the $x$-axis and below the circle of centers $C_1(0,1)$ and $C_2(2,1)$, both of radio equal to $1$. $C_1: x^2+(y-1)^2=1$ and $C_2: (x-2)^2+(y-1)^2=1$...
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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 ...
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0answers
30 views

Proof of Green's theorem under assumption of only type 1 curve.

The proof of Green's theorem on Stewart's calculus book has assumption of the closed curve being both type 1 and type 2. Can one prove the theorem under assumption of being only type 1? Are there ...
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1answer
32 views

integral of a function in a curve

let $F=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ and $R(t)=(\cos t,\sin t)$ (the curve is a circle with radius 1) now: \begin{equation} \int_{R}F_1.dx+F_2.dy = \int_{0}^{2\pi}-\sin t\ dt + \...
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1answer
63 views

Use Green's Theorem to evaluate a line integral

Evaluate the line integral $\int_cy^4\ dx+2xy^3\ dy$ where $C$ is the ellipse $x^2+2y^2=2$. My attempt: First, I need Green's Theorem: $\int_cP\ dx+Q\ dy = \int\int_D\big(\frac{\partial{Q}}{\...
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0answers
46 views

Stuck in a Calculus exercise

I'm studying for my exams coming up next week. The exercise is the following: Let $\Omega \subset \mathbb R^2$ be a simply connected space of area equivalent to $2.$ Let $\gamma $ be a simple ...
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4answers
256 views

Is the solution of $ u_{tt}=c^2u_{xx}+xt $ correct?

Consider the following $ u_{tt}=c^2u_{xx}+xt,\\ u(x,0)=0,\\ u_t(x,0)=\sin (x)$ and find the solution. Solution. We have that $u(x,t)$ is given by $$u(x,t)=\frac{1}{2}(g(x+ct)+g(x-ct))+\frac{1}{2c}\...
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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 ...
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1answer
21 views

Calculation of Line-integral: parametrization problem

Calculate $$\int_Cxy^2\ \text{d}y-yx^2\ \text{d}x$$$C=\{(x,y)\in\mathbb R^2:x^2+(y-1)^2=1\}$ using Green's theorem: $\int_{C}P\ \text{d}x+Q\ \text{d}y=\int_D(\frac{\partial Q}{\partial y}-\frac{\...
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2answers
52 views

Parametrisation of this triangle

I have been doing an exercise problem, and there is something unclear about it. Namely, how was the triangle in the following example transformed to a circle $C'$, which is parametrised by $(x(t), y(t)...
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1answer
45 views

Is this statement true?: $\frac{\partial{F1}}{\partial{y}}=\frac{\partial{F2}}{\partial{x}}$ for some smooth $\vec{F}\iff \vec{F}$ is conservative

Here is a statement about conservative vector fields to be proved or disproved: If $\vec{F}=(F1,F2) :\Omega \rightarrow \mathbb{R}^2\space$is a smooth vector field (here $\Omega\subseteq\mathbb{R}^2$)...
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0answers
22 views

Verification of Green's Theorem in a Simple Case

Show that Green's Theorem is satisfied for $\vec{G}: \mathbb{R}^2\rightarrow\mathbb{R}^2$ $\vec{G}=y\vec{i}-x\vec{j}$ and the path $C: x=\sin t, y=\cos t$ for $0\leq t \leq 2\pi$. We want to show $$\...
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2answers
61 views

Show $\oint_C (Pdx+Qdy) \,=2\pi$ if $P$ and $Q$ aren't $C^1$

Define $P=\frac{-y}{x^2+y^2}$ and $Q=\frac{x}{x^2+y^2}$ for $(x,y)\neq(0,0)$ I wish to show that $$\oint_C (Pdx+Qdy) \,=2\pi \\$$ where C is any circle with centre $(0,0)$ orientated counterclockwise....
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1answer
39 views

Multivariable Calculus, Help with pre-Bachelos Homework [closed]

I need help with a couple of problems for homework, btw I need to finish this in less than 12 hours, I'll appreciate your help, and srry if something is wrong or is confusing I'm not native speaker. ...
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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):...
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1answer
39 views

Line integral using greens theorem and polar coordinates

The vectorfield F is given by $$\mathbf{F}(x,y) = (x^3 - y, x + y^3) $$ Calculate $$ \oint_C \mathbf{F} \cdot d\mathbf{r}$$ Where C is the boundary of the region enclosed by $y = x$ and $y = x^2$ and ...
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1answer
31 views

Equation for area of closed simple curve

Suppose we have a closed simple curve written as $(x,y) = F(u)$. To find the area enclosed by this curve, we appeal to Green's theorem: $$ A = \frac{1}{2}\int_0^{2\pi} [x\frac{\partial y}{\partial u} ...