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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30 views

Two multivariable integral questions.

Let $\mathbf{F}(x, y) = (-y^2, xy)$ and $C = \Bigl\{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 : y \ge 0 \Bigr\}$. Determine $\displaystyle \int_C \mathbf{F} \cdot d\mathbf{x}$ if $C$ is oriented counter ...
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Verify step in proving Green's theorem (oriented 2-cells)

One of the step in proving Green's theorem is:(from Advanced Calculus of Several Variables) For every nice region that is the image of unit square $I^2$ under a suitable mapping. The set $D$ subset in ...
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230 views

How to solve this Green's theorem Question?

I have been given the question by my university as an Assignment to solve. I have solved but when I shared with my fellows they are saying the answer of this question is zero, but I am getting the ...
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Well define of Green's theorem $\iint\limits_D d\omega =\int_{Fr(D)} \omega $

$$\iint\limits_D d\omega =\int_{Fr(D)} \omega $$ Where $\omega$ is differental form such as $\omega=Pdx+Qdy$ and $d\omega=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$ Let $Fr(D)$ be a ...
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Software to approximate area of curve

Does any know of a math program where I can measure the area of a closed parametric curve ? I know that I can measure the area between 2 curves with the TI-Nspire, but not for one curve in ...
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1answer
47 views

$\int_\gamma{(x-y)dx + (x+y)dy}, \quad \gamma : x^2 + 2y^2 = 1 , \quad 0 \leq y $

I'm asked to find $$\int_\gamma{(x-y)dx + (x+y)dy}$$ where $$\gamma : x^2 + 2y^2 = 1 , \quad 0 \leq y$$ (with positive direction) i.e the upper half of the ellipse $x^2 + 2y^2 = 1$. My attempt ...
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1answer
108 views

understanding Green function, Boundary element method, Green element method from scratch

I am newly exposed to Green function, Boundary element method, Green element method and would like to understand them from scratch in solving Parabolic PDES (specifically flow In heterogeneous ...
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1answer
68 views

Prove $\int_0^{2\pi} \sin^{2n}t dt=\frac{2n-1}{2n}\int_0^{2\pi}\sin^{2n-2}tdt$

Prove $$\int_0^{2\pi} \sin^{2n}t dt=\frac{2n-1}{2n}\int_0^{2\pi}\sin^{2n-2}tdt$$ for all integer $n>0$. My attempt: Let $x=\cos t$, $y=\sin t$. Then $\sin^{2n}tdt=y^{2n}(-\frac{dx}{y})$ since $dx=-...
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43 views

How can the following integral be evaluated without using Green's Theorem?

How can the following integral be solved without using Green's Theorem and without converting it into a line integral? $\iint_{R}(-1)dxdy$ where R is the region enclosed by $x=\cos(t)$, $y=2\sin(t)$, ...
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Problems with final result of planimeter proof

I am referring to this explanation of the polar planimeter, pp. 7 I think I do understand the following result: During this operation, the wheel on the tracer arm will cover a distance of ...
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219 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(\...
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1answer
292 views

Green's Theorem in polar coordinates - what am I doing wrong?

Let $$\vec F = M\hat i + N\hat j = (3x^2y^2)\hat i + (2x^2 + 2x^3y)\hat j$$ be a vector field, and C be the counterclockwise circle centered at $(a,0)$, with a radius of $a$. Find $\oint_C M\,dx + N\,...
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151 views

Calculate $ \oint_C y^2\, dx + x\, dy $ using Green's Theorem?

Let $C$ be the curve parametrized by the equation $r(t) = 2\cos^3(t) i + 2\sin^3(t) j$ for $t \in [0,2\pi]$. I want to find the line integral $$ \oint_C y^2 \,dx + x \,dy . $$ I evaluated it ...
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50 views

Area of parametrized curve

A simple closed curve $C$ is given by $\textbf{x} = (f(t), tf(t)), t\in [a,b]$. Show that the area enclosed by the curve is given by $A = \frac{1}{2}\int_{a}^{b}f(t)^2dt$. I tried to use Green's ...
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451 views

Using Green's Theorem when the function is not defined at the origin.

I need to use Green's theorem to calculate the following integral over the curve $C = \frac{x^2}{16} + \frac{y^2}{9} = 1$, orientated with the hands of a clock: $$ \int_{C}\frac{y^3dx-xy^2dy}{(x^2+y^2)...
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673 views

Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent ...
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270 views

vector calculus book for Green's theorem

I am finding good vector calculus books for rigorous proof of Green's theorem since I have learnt cauchy's theorem in complex analysis can be proved so easily. Can anyone recommend?? +: is calculus ...
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1answer
739 views

the 2-D divergence theorem and Green's Theorem

I read somewhere that the 2-D Divergence Theorem is the same as the Green's Theorem. So for Green's theorem $$\oint_{\partial \Omega} {\textbf{F}} \cdot d {\textbf{S}} = \iint_{\Omega} \text{2d-curl}{\...
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Green's theorem, double integral over a triangle

The original problem is given as thus Find $$\iint_Dx\,dxdy $$ where $D$ is a triangle with vertices $(0,2),(2,0),(3,3)$. Green's theorem says that $$\iint_D(G_x-F_y)dxdy = \int_{\partial D}Fdx+...
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vector integration

Question- Evaluate by Green's theorem for $$\oint_C\frac{1}{y} dx + \frac{1}{x}dy$$where C is boundary of the region defined by $x=1, x=4, y=1, y^2=x.$ I solved this problem and got $-27/4$ as the ...
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Are single points or rectangles of area $0$ simple closed curves?

I was solving the following problem: Show that $\oint_C{-x^2ydx+xy^2dy} > 0$ around any simple closed curve $C$. I began with applying Green's Theorem: $\oint_C{-x^2ydx+xy^2dy} = \iint_R{y^2+x^...
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343 views

Applications of Green's Theorem for which there are no obvious simpler proofs

Green's Theorem is a neat theorem in that it relates a double integral over a region in the plane to an integral of a vector field on its boundary. One of my favorite applications is using it to find ...
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Applying Green's Theorem to evaluate line integral

"Apply Green's Theorem to evaluate the line integral of F around positively oriented boundary" $$F(x,y)=x^2yi+xyj$$ C: The region bounded by y=$x^2$ and y=4x+5
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1answer
355 views

Greens theorem over a trapezoid

This is the solution to a problem on greens theorem bounded by a trapezoid. I am stuck on the third last equality sign. I suspect it has to do with symmetry of the domain but can not see how it has ...
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101 views

Apply Green's theorem for $\iint_D (x+2y)dxdy$ on region bounded with $x = t- \sin (t), y = 1 - \cos (t)$.

Basically, I am stuck with his exercise. It asks to compute the integral: $$\int_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dxdy = \iint_D ...
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1answer
805 views

Green's Theorem for area using polar coordinates

I've come across a Green's Theorem proof that has me perplexed. Using the area formula: $$A = \frac{1}{2}\int_C xdy - ydx $$ Prove that:$$A = \frac{1}{2}\int_a^b r^2d\theta$$for a region in polar ...
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1answer
80 views

Green's theorem on a trigonometric curve

I'm reviewing for an exam, and picked this problem out for practice: Use Green's theorem to evaluate $$\oint_C y^2dx+xdy \,$$ For the curve $C$, $$C = \alpha(t) = 2\cos^3(t) \hat{i} + 2\sin^3(t) \...
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72 views

Prove for any closed curve $C$ then the path integral is independent of the path between points $p$ and $q$

Prove that if $$\oint_C \vec{B}.d\vec{r}=0$$ for any closed curve $C$, then the path intergral $$\int_P^Q\vec{B}.d\vec{r}$$ is independent of the path taken between points $P$ and $Q$. Any help would ...
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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'...
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1answer
109 views

Work done in a Vector Field / Green's Theorem with Parametrization

I have a question that states: Calculate the work done by the vector field $$F = y^3 \hat{i} + 3xy^2 \hat{j}$$ A parametrized version is given as $$x(t) = cos(t) +\frac{1}{4}sin(5t)^2$$ $$y(t) = sin(...
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1answer
742 views

How to calculate the area of an asteroid using Green's theorem

I came across this question in my revision: Use Green's theorem to calculate the area of an asteroid defined by $x=\cos^3{t}$ and $y=\sin^3{t}$ where $0\leqslant t \leqslant 2\pi$ . The question gives ...
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1answer
208 views

How to use Green's theorem?

$\def\d{\mathrm{d}}$I'm thinking about this differential equation $$\frac{3}{2} x\,\d x + \frac{x}{y}\,\d y = 0.$$ If functions $P(x,y), Q(x,y)$ are difined as$$P = \frac{3}{2} x,\ Q = \frac{x}{y},$$...
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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 ...
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1answer
159 views

Rellich Integral Identity for Normal Derivatives

Consider the differential operator on $u \in C^2(\overline \Omega)$ with $u \bigr|_{\partial \Omega}=0$ and $\Omega \subseteq \mathbb{R}^n$ $$ Au:= x \cdot \nabla u $$ Then I have seen the following "...
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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}\;/\;...
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1answer
160 views

Region of ellipse and greens theorem

Let γ be the ellipse $x^2 + 4y^ 2 = 4$, oriented anticlockwise. Compute $\int_c(4y − 3x)dx + (x − 4y)dy$ I used green theorem with P and Q. and got $$-3\int\int_\ dxdy$$ The answer is $-6\pi,$ so ...
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86 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 ...
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330 views

When Green's Theorem Creates An Integral Of $0$, What Happens?

"Evaluate the counter-clockwise integral: $\int_c 4x^3ydx + x^4dy$ for any closed path $C$." My Work This is obviously a job for Green's theorem, where $P = 4x^3y$ and $Q = x^4$ $\frac{\partial Q}...
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97 views

Are holomorphic functions divergence free?

I am attempting to understand a proof in a paper I'm reading. I am stuck on the first line. We construct a box, R, in the complex plane. Then, by Green's theorem, we have $$\int_{\partial R}\frac{\...
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1answer
809 views

Green's formula in n-dimensions?

I'll try to get straight to the point. The professor teaching my FEM course is a bit of a stereotype professor in that he is quite hard to understand. Green's formula is used quite extensively in ...
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155 views

Calculate area bounded by curve using Green's theorem

For homework, I need to find the area bounded by the curve $x^3+y^3=3xy$. We were given a series of hints, of which I understand very little. First, we're told to set $y=tx$ to obtain a ...
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Does the potential exist?

Verify Green's theorem for $X(x,y)=(xy^2,-yx^2)$ in the circle of radius $R$ with center $(0,0)$. I think there is a mistakein the field. I suppose the first thing I should do is to find a function $...
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1answer
354 views

Green's Theorem in the Unit Square

An exercise in a book says "Prove Green's theorem for $R=[0,1]\times[0,1]$". It doesn't add any specific form, so I will assume it askes to prove $\int_R d\phi=\int_{\partial R} \phi$ for an arbitrary ...
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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 ...
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1answer
29 views

Integration by parts of non-parametric partial derivative

Let $A(x,y,z)$ be some smooth, but wiggly function on $R^3$. Let $B(x,y,z) = \frac{\partial A}{\partial z}$. Let $x,y \in\Gamma$, which is a simple 2D surface. In my case, it is a planar triangle. I ...
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1answer
31 views

Work on a line segment - Final step

I'm trying to solve the following problem: Compute the work of the vector field $F(x,y)=(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2})$ in the line segment that goes from (0,1) to (1,0). My attempt (...
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1answer
64 views

Integrating vector function

Integrate the vector function $$F=(2x)\hat{i}+(4y)\hat{j}-(5z)\hat{k}$$ Over the closed surfaces of the volume defined between the surfaces of $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=1$ and $z>0$. I ...
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69 views

Green's theorem details

I'm studying Green theorem. My textbook gives me two examples in which I have a few doubts 1st example Imagine the annulus $D=\{(x,y) \in \mathbb{R^2}: r^2 < x^2 + y^2 < R^2\}$ $F$ ...
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1answer
473 views

Prove Identity for Biharmonic Operator using Green's Identities

Prove that for every $u\in \mathcal C^2(\bar \Omega)$ where $\Omega$ is an open and bounded subset of $\mathbb R^n$ and $\partial \Omega \in C^1\;$ the following holds for every $y\in \Omega$: ...
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1answer
70 views

Show that $L(v) = \int_{\Gamma} gv ds$ is a continuous operator

This is problem 2.4 from "Numerical Solution of Partial Differential Equations by the Finite Element Method" by Claes Johnson. Let $\Omega$ be a square with boundary $\Gamma$. Show that there is a ...