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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39
votes
3answers
3k views

Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$ The Cauchy-Riemann equations have the ...
11
votes
2answers
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 ...
9
votes
3answers
201 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...
7
votes
1answer
612 views

How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
7
votes
1answer
473 views

Differential equation - Green's Theorem

I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ ...
6
votes
1answer
249 views

Using Green's Theorem to Express the Integral $I=\int_C (Pdx+Qdy)$ as an expression of $I_i=\int _{C_i} (Pdx+Qdy)$

Let $p_1,...p_n$ be points in $\mathbb{R}^n$. Let $P(x,y), Q(x,y)$ be functions with continuous derivatives in $ D=\mathbb{R}^2\setminus\{p_1,...p_n\}$ such that $Q_x-P_y=1$ for all $(x,y)\in D$. For ...
5
votes
1answer
303 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
4
votes
2answers
513 views

Double Integral Formula for Holomorphic Function on the Unit Disc (Complex Plain)

While studying some complex analysis , I encountered the following problem: "Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$.Prove that for every $\zeta \in \mathbb{D}$ the ...
4
votes
2answers
800 views

Solving line integral without using Green’s theorem

Good morning everyone, I am a new user here and I have the following problem that I am dealing with since yesterday but I cannot find the correct answer. Here is the problem Calculate $$\oint_C (x^2-...
4
votes
1answer
206 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$$P = \frac{3}{2} x,\ Q = \frac{x}{y},$$...
4
votes
0answers
120 views

Gaussian integral over a wedge of the complex plane

I'm trying to evaluate, if it is possible, a Gaussian integral over a wedge of the complex plane $$ I = \int_W d^2z \,e^{-|z|^2 + z^*a + za^*}, $$ ($d^2z = \frac{dz\,dz^*}{2} = dx\,dy = r\,dr\,d\...
4
votes
1answer
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 ...
4
votes
0answers
178 views

Green's first identity and the calculus of variations

UPDATE: I was able to solve this problem using iterative integration by parts. However, I still cannot find how Green's first identity would apply here. Suppose I had a multiple integral over $p$-...
3
votes
1answer
35 views

Solve $4\int_0^a\int_0^{\sqrt{a^2-x^2}}\int_0^b5x^2dxdydz$

I tried solving this problem: Evaluate $\iint_Sx^3dydz+x^2ydzdx+x^2zdxdy$ Where S is the surface bounded by $z=0, z=b, x^2+y^2=a^2$ Using Green Theorem, $\iint_Sx^3dydz+x^2ydzdx+x^2zdxdy=\iiint_V\...
3
votes
2answers
61 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 ...
3
votes
2answers
347 views

Greens theorem on the circle $x^2 + y^2 = 16$.

Use Greens theorem to calculate the area enclosed by the circle $x^2 + y^2 = 16$. I'm confused on which part is $P$ and which part is $Q$ to use in the following equation $$\iint\left(\frac{\partial ...
3
votes
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 ...
3
votes
1answer
1k views

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+...
3
votes
2answers
110 views

$f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|\leq 2\}$ such that $\iint_D=|f(z)|^2\,dx\,dy\leq 3\pi$. Maximize $|f''(0)|$

Determine the largest possible value of $|f''(0)|$ when $f$ is an analytic function in the disk $D=\{z\in\mathbb{C}\,:\,|z|<2\}$ with the property that $\iint_{D}|f(z)|^2\,dx\,dy\leq 3\pi$. I don'...
3
votes
1answer
53 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 ...
3
votes
0answers
52 views

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'...
3
votes
0answers
83 views

Aproximation to Jordan curve by polygonal jordan curve

In the proof of the Green theorem (in the last step: any region can be approximated as closely as we want by a sum of rectangles), I need to prove the following result: Let $\gamma$ be a Jordan curve ...
3
votes
0answers
88 views

Checking the validity of Green function using a operator using a certain limit

We have an operator of green function which can be written as $\Big(-i \frac{\partial}{\partial x} -a \Big)^2$ within the periodic boundary conditions, where $x,y \epsilon ]-\pi, \pi].$ Using ...
3
votes
0answers
214 views

Green's theorem application in Complex analysis

Let $\phi\in C_c^{\infty}(\mathbb{C}).$ Prove that $\displaystyle \int_{|z-w|>\epsilon} \log|z-w|\Delta\phi(z)dA(z)=\int_0^{2\pi}(\phi(w+re^{it})-r\log r\frac{\partial \phi}{\partial r}(w+re^{it})|...
3
votes
1answer
88 views

How does Green's theorem apply here?

Let $D$ be the region delimited by $$\partial D: \begin{cases} C_1: x^2 + y^2 = 5^2\\ C_2:(x-2)^2+y^2= 1\\ C_3:(x+2)^2+y^2 = 1\\ C_4: x^2+(y-2)^2= 1\\ C_5: x^2+(y+2)^2= 1 \end{cases} $$ I've sketched ...
3
votes
2answers
131 views

Green's Theorem on Line Integral

I am asked to find the line integral for the following field: $$F = (e^{y^2}-2y)i + (2xye^{y^2}+\sin(y^2))j$$ On the line segment with points $(0,0),(1,2)$ and $(3,0)$. I have to do it with Greens ...
2
votes
3answers
480 views

Use greens theorem to find work done

Use Green's Theorem to find the work done by the force $\mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j}$ in moving a particle from the origin along the $x$-axis to $(1,0)$, then along the line segment ...
2
votes
2answers
6k views

Proving Green's Theorem for Computing Area

I'm having a hard time understanding where exactly the formula for computing area using Green's theorem comes from. It is typically: \begin{equation} \int_C x\,dy = \int_C -y\,dx = Area \end{...
2
votes
4answers
241 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}\...
2
votes
2answers
45 views

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 $...
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
31 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 + \...
2
votes
1answer
738 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}{\...
2
votes
1answer
80 views

Prove that $\lim_{\epsilon \rightarrow 0}\int_{\partial B_\epsilon} (φ∇g · n − g∇φ · n) ds = 2πφ(0, 0)$

Suppose $φ : \mathbb{R^2}\rightarrow \mathbb{R}$ is any $C^1$ function and let $g:\mathbb{R^2}-\{(0,0)\}\rightarrow \mathbb{R}$ given by $g(x, y) := \ln\sqrt{x^2+y^2}$ Prove that $\lim_{\...
2
votes
1answer
7k views

What is a x-simple and y-simple region? [closed]

I have read the definition, but I do not quite understand what it means.
2
votes
1answer
38 views

Why does the boundary of a region $D$ have enough information to dictate the value of an integral over $D$?

There are many theorems which say something along the lines of the title: The FTC: $\int_a^bf'(x)dx=f(b)-f(a)$. Green's Theorem: Let $F=(P,Q)$, then $\oint_{\partial D}Fds=\iint_D(\frac {\partial Q}{\...
2
votes
1answer
685 views

Does Greens Theorem apply to the annulus?

I realize that it does, but I can't prove it. If the region is of the form: $$ D = \{ (x,y) ) \ | \ x \in [a,b] , \ \mu(x) \le y \le v(x) \}$$ or, with $y$ and $x$ changing place (and especially if ...
2
votes
1answer
249 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 ...
2
votes
1answer
56 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}}{\...
2
votes
1answer
222 views

Applying Green's theorem on scalar fields

Consider the vector field $\mathbf F = \langle x+x^3e^y, -3x^2e^y \rangle$ and let $C$ be the circle $x^2 +y^2=5^2$. Let $\mathbf u$ be the unit vector $\dfrac{1}{\sqrt{x^2+y^2}}\langle x,y\rangle$ ...
2
votes
1answer
100 views

Showing the Law of Conservation of Mass is equivalent to the continuity equation?

Let $\textbf{v}(t,x,y,z)$ be a continuously differentiable vector field over the region $D$ in space and let $p(t,x,y,z)$ be a continuously differentiable scalar function. The variable $t$ represents ...
2
votes
1answer
46 views

Applicability of gradient theorem in the calculation of flux.

I have a small confusion with the applicability of the fundamental theorem of line integral. The theorem states that if $F=P\vec{i}+Q\vec{j}\;$ is a gradient field ($\nabla g$) of some function $g$ ...
2
votes
2answers
1k views

Green's Theorem for 3 dimensions

I'm reading Introduction to Fourier Optics - J. Goodman and got to this statements which is referred to as Green's Theorem: Let $U(P)$ and $G(P)$ be any two complex-valued functions of position, ...
2
votes
1answer
241 views

Why is a semiannular region not simply connected?

In a textbook I am consulting re: Green's Theorem, they illustrate with the following example: Evaluate $\int_C y^2\, \text{d}x + 3 xy \, \text{d}y$, where $C$ is the boundary of the semiannular ...
2
votes
5answers
187 views

Evaluate the contour integral (Most likely without Green's Theorem)

$\int_{c}\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ where $C$ is the triangle with vertices at $(5,5), (-5,5),$ and $(0,-5)$ traversed counterclockwise. (Hint: Be careful about the hypotheses of any ...
2
votes
1answer
482 views

Green's Theorem with change of variables

Evaluate $$\int_C F dr$$ $$ F =< x2, xy > $$ $$ C: x^2/4^2 + y^2/9 = 1 $$ With y ≥ 0 positively oriented. For the circle $$ u^2 + v^2 = 1 $$ $$ u=x/a $$ $$ v=y/b $$ $$ x=au $$ $$ y=bv $$ ...
2
votes
1answer
4k views

Green's Theorem for an off-centered circle

I have the following problem where I'm trying to figure out how to convert a circle whose equation is $(x-1)^2 + (y+3)^2 = 25 $ traversed counterclockwise. Here's the integral $$ \oint_C 2ydx + 5xdy ...
2
votes
1answer
37 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}},\...
2
votes
1answer
73 views

Calculate flux integral

Let N be the upwards unit normal on the surface $M:z=e^{-x^{2}-y^{2}},x^{2}+y^{2}\leq1$. Consider the vector field u=$(x^{4}-y^{3},cos(x),sin(z))$ Im asked to compute the flux integral $\int \int_{M}...
2
votes
1answer
756 views

2D Divergence Theorem: Question on the integral over the boundary curve

Let $\;F=(F_1,F_2)\;$ be a two-dimensional vector field and consider the rectangle $\;\mathcal R= PQRS\;$: If $\;\vec v\;$ is a function which gives outward-facing unit normal vectors to $\;\...