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Questions tagged [greens-theorem]

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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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$\int_{C}^{}v dS$, closed curve $C$, stable array $v$

I have a theoritical question from line integrals. If the closed curve $C$ is the border of surface $S$, and $v$ is a stable array, then what can i say about $\int_{C}^{}v dS$? My thoughts: Can I use ...
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38 views

Green's identity formula

Instead, if we assume that $v\in C^\infty_c(U), u\in C^2(U)$. Will the second equality still hold? Since $v$ have compact support in U, the boundary term will just vanish and $-\int_Uu\Delta v dx$ ...
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1answer
59 views

Green's Theorem and Line Integrals

So I'm supposed to use Green' theorem to calculate the line integral $$ \int_{C_1} \frac{x^2-1}{x^2+4y^2}dx +\frac{x}{x^2+4y^2}dy $$ Where $C_1$ is the part of the parabola $y=1-x^2$ from point $(1,0)$...
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18 views

Can we assure $\frac{d^2}{dx^2} G(\textbf{r},\textbf{r}') = \frac{1}{3} \delta (\textbf{r}-\textbf{r}')$?

Recently, I've made a question regarding the proof of $\nabla ^2 G(\textbf{r},\textbf{r}') = \delta (\textbf{r}-\textbf{r}')$ for $G(\textbf{r},\textbf{r}')=\frac{1}{4\pi|\textbf{r}-\textbf{r}'|}$. ...
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81 views

Confusion regarding the solution to $\nabla ^2 \phi(\textbf{r}) = \rho(\textbf{r})$ using Green's function [duplicate]

We know we can solve one of the Maxwell's equation using Green's function. More specifically, we can solve $$\nabla ^2 \phi(\textbf{r}) = \rho(\textbf{r})$$ using $$\phi(\textbf{r}) = \int d\textbf{r}'...
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40 views

Why is Green's theorem not valid for $\{(x, y):x^2 +y^2 <= 1\}\setminus \{0\}$ for $F=- y/(x^2+y^2)\vec{i} - x/(x^2-y^2)\vec{j}$

Why is Green's theorem not valid for $$\{(x, y):x^2 +y^2 \le 1\} \setminus \{0\}$$ for $$\vec{F}=\frac{-y\vec{i}}{x^2+y^2} - \frac{x\vec{j}}{x^2-y^2} $$ Clearly $dF/dx=dF/dZ$ then why is the integral ...
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1answer
27 views

Can we use Green's Theorem to find line integral where circles have different centres?

Suppose we have to find a line integral where $C$ is the boundaries of two circles with different centres. Can we use Green's Theorem here? Since centres are different its difficult for me to convert ...
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Gauss-Ostrogradsky's theorem

Question: Is there any formula that bounds the line and double integrals other than the Green one? My guess: No! We know: $$ \int_V \operatorname{div} \vec{F}\, dx\,dy\,dz = \int_{\partial V} \vec{F} \...
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40 views

Line integral under closed sign

Compute $$ \oint_{C}\left[\left(2x - y^{3}\right)\mathrm{d}x - xy\,\mathrm{d}y\right], $$ where $C$ is the boundary of region enclosed by $x^{2} + y^{2} = 1$ and $x^{2} + y^{2} = 9$. I am confused ...
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16 views

For what simple closed curve $C$ does the line integral of $f(x, y)$ achieve its minimum value? Using Green's Theorem.

Over a curve $C$ the line integral $$\int \limits_C [-(x^2)y - 3x + 2y]dx + [4(y^2)x-2x]dy.$$ Applying Green's Theorem -> Over a region $R$ $$\iint \limits_R [x^2 + 4y^2 - 4]dx dy$$ Answer: On the ...
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1answer
65 views

mollification of harmonic function is harmonic (Green's identity)

For the equality $\int_{\mathbb{R}^n}\Delta_y \eta_\epsilon(x-y)u(y)=\int_{\mathbb{R}^n} \eta_\epsilon(x-y)\Delta_yu(y)$, did we perform integral by parts twice? If so, do we need any information of u ...
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1answer
43 views

Compute area with Green's Theorem

$\def\hl#1#2{\bbox[#1,1px]{#2}} \def\box#1#2#3#4#5{\color{#2}{\bbox[0px, border: 2px solid #2]{\hl{#3}{\color{white}{\color{#3}{\boxed{\underline{\large\color{#1}{\text{#4}}}\\\color{#1}{#5}\\}}}}}}} \...
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42 views

Minimizing a line integral

Let $F:\mathbb{R}^2 \to \mathbb{R}^2 $ s.t $$ F(x,y) =\begin{bmatrix} x^2 +y^2 +4y \\ x^3 +xy^2 +2xy\end{bmatrix} $$ Find the simple closed curve that minimizes $\int_C F \cdot d\mathbb{x}$ I've ...
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72 views

Compute $\int xy dx +(x+y)dy$ over the curve $Ī“$, $Ī“$ is the arc $AB$ in the 1st quadrant of the unit circle $x^2+y^2=1$ from $A(1,0)$ to $B(0,1)$.

Compute $\int xydx+(x+y)dy$ over the curve $Ī“$, where $Ī“$ is the arc $AB$ in the first quadrant of the unit circle $x^2+y^2=1$ from $A(1,0)$ to $B(0,1)$. I solved this problem with the help of Green'...
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calculating Area inside intersection of circle and ellipse using line integral

Consider a circle parametrized as $(r\cos (t), r \sin (t))$ and an ellipse parametrized as $(a\cos (t), b \sin (t))$. Assuming that $a>r>b$, you find the area of region of intersection of circle ...
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32 views

Line integral of an ellipse using Green's theorem

Let C be the curve in $\mathbb{R}^2$ defined by, C: $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1, x\geq0,y\geq0$ Compute the line integral, in a direction so that the $y$-coordinate increases: $\int_C (2x-3y)dx+(...
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1answer
43 views

How to calculate line integral using Green's theorem

I had this specific task in my math exam and didn't solve it correctly. Also, I, unfortunately, don't have any correct result. So I am asking you, if anyone can solve and explain it to me. I would be ...
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Understanding Complex Form of Green's Theorem

I'm reviewing complex analysis for the GRE. I've never taken a course in complex analysis before, but I do know vector calculus. I'm trying to understand the statement of the complex version of Green'...
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1answer
32 views

Can a vortex vector field be conservative?

For the following vortex vector field $$F(x,y)=\left(\frac{2xy}{(x^2+y^2)^2},\frac{y^2-x^2}{(x^2+y^2)^2}\right)$$ If we apply the extended Green's Theorem for an arbitrary simple closed curve $C$ that ...
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4answers
62 views

Elementary calculus, looking for different approach

We want to calculate the area of the shape $S$ bounded by the curve $\gamma$ defined by $\begin{cases}x = a\cos^3(\theta) \\ y = a\sin^3(\theta)\end{cases}$, where $a \in \mathbb R$ and $0 \leq \theta ...
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219 views

Reference for proof of Green's theorem

I'm looking for a rigorous proof of Greens theorem for piecewise smooth jordan curves and would appreciate if someone could link a reference text. The only proof I've seen works for regions which can ...
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1answer
52 views

Integration by parts with cross derivatives

I wish to solve the following simplified problem in the context of Weak Formulations $\large \iint(u \frac{\partial ^2 v}{\partial x ^2})dxdy + \iint(u \frac{\partial ^2 v}{\partial x \partial y})dxdy ...
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57 views

Find the Work $\oint Pdx+Qdy$ of the $F(x,y)=(āˆ’y^2,x^2)$ on the rectangle $xāˆ’y=0$,$xāˆ’y=1$,$x+y=1$,$x+y=2$ (positive direction)

Find the Work $\oint{Pdx+Qdy}$ of the $F(x,y)= (-y^2,x^2)$ on the rectangle (pic) $x-y=0, x-y=1, x+y=1, x+y=2$ (positive direction) From Green's theorem: $$\oint{Pdx+Qdy}= \iint{\left(\frac{\partial ...
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34 views

Applying Green's Theorem Question

We've just been introduced to Green's Theorem and I've been working on some homework questions and came across one which I have now been struggling with for a couple of days. And help would be ...
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How to come up with Dulác-Function?

I'm currently studying dynamical systems and came across the Bendixson-Dulac-Theorem Let $D \subseteq \mathbb{R}^2$ open, $f \in \mathcal{C}^1(D, \mathbb{R}^2)$ and consider the nonlinear system $...
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22 views

Where to find reference for the energy method $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm}$

Why do I need to multiply by the function w in the energy method to guaranty at most one solution? This is the example $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm} u=0 \hspace{0....
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1answer
52 views

How to Prove a Special Case of Stokes' Theorem?

I am currently in Calculus 3, or Multivariable Calculus and need to prove this special case of Stokes' theorem. Please forgive me as I do need this simplified to the bones to understand the ...
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1answer
36 views

How is obtained the formula $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx $ [closed]

How to show that this inequality holds? $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx $ Where $\psi$ is a scalar function and $F$ ...
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1answer
34 views

Why this does not have a dot product on the right: $\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$

My main concern is because, on the right side of the expression, there is a vector $\nu$, but there is not any dot product. $$\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$$ Where $\...
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Vector calculus - Green's Theorem image

I'm working on the circulation form of Green's Theorem in a math textbook and came across this image. Of course, I understand r is the parametrization of C but the figure has what look like non-...
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24 views

Sign of Divergence Implying Zero Net Outward Flux

Suppose you have a simple closed curve C bounding a region R. The region R has negative divergence, and is enclosed by another region with positive divergence such that on the closed curve C, the ...
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1answer
36 views

Green's Theorem to calculate a line integral

Use Green's Theorem to evaluate the line integral $\int_C y^3dx + x^3dy$ where $C$ is the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$, so I did this using symmetry. However, I friend switched to ...
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1answer
25 views

Divergence Theorem Clarification

This is the first time I'm doing this kind of problem, and I'm (understandably I think :)) hesitant about my answer. The question is: Compute $\iint_{\partial Q}\ F\bullet ndS$ using the easiest ...
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1answer
37 views

Computing integrals

Compute $I := \int_Df(x, y)$ with $f(x, y) := 2y \sin (x)$ and $D$ rectangle with vertices $A := (0, 0)$, $B := (Ļ€, 0)$, $C := (Ļ€, 1)$ and $D := (0, 1)$. Compute $I := \int_D \sqrt{x^2 + y^2}$ with $D ...
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26 views

how can I conclude $\|v\|^2_{H^2(\Omega)} \leq C \int (\Delta v)^2 dx$?

I need to prove that : Let $\Omega$ be a square with boundary $\Gamma$. Show that there is a constant C such that $\|v\|^2_{H^2(\Omega)} \leq C \int_\Omega (\Delta v)Ā² dx $ $\forall x \in H^2_0(\...
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1answer
20 views

Definition of generalized Laplacian by Green's theorem

I am reading a book on Potential Theory and they motivate the generalized Laplacian by saying that Green's theorem implies the following: $$ \int_D \phi \Delta u = \int_D u \Delta \phi dA $$ where $\...
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1answer
152 views

Green's theorem (integration by parts) on the unit sphere

List item I am missing something here and I need help to find it: Since the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ has no boundary, then given a smooth function $\phi$ and a smooth ...
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If $\Omega=\{\vec{x}\in U| f(\vec{x})\not = 0\}$ then $\int_{\Omega}\frac{\partial f}{\partial x}=0=\int_{\Omega}\frac{\partial f}{\partial y}$

I'm having problems with this proof. I will appreciate any hint. Given $f: U\subset \mathbb{R}^{2}\mapsto \mathbb{R}$ of class $C^{1}$ in the region $U$. If $\Omega=\{\vec{x}\in U| f(\vec{x})\not = 0\}...
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41 views

Complex integral to determine area inside of parameterized closed curve

By combining several things I read in several places, it seems that the area inside a parameterized closed curve in the complex plane ($\;f : [0,1] \to \mathbb C\;$ with $\;f(0)=f(1)\;$ and 'piecewise ...
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2answers
55 views

Invariance of line integral

I need to determine functions $P,Q\in C^2(\mathbb R^2)$ provided that the line integral \begin{equation} I=\int_LP(x+\alpha,y+\beta)dx+Q(x+\alpha,y+\beta)dy \end{equation} over closed curve $L$ doesn'...
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Calculate the line integrals of these conservative functions

$F := (\frac{x}{x^2+y^2+1} ,\frac{y}{x^2+y^2+1})$ and $Ī“ := (x, y) : x^2 + y^2 āˆ’ 2x = 1$ , $Ī“$ traversed counterclockwise $F := (2xy^2z, 2x^2yz, x^2y^2 - 2z)$ and $Ī“ :=$ {$(cos(t),\frac{\sqrt{3}}{2} ...
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1answer
27 views

Green's theorem in comparison to normal line integrals

I am having trouble internalizing the geometric significance of Green's Theorem and Line Integrals in general. Why is it that a normal line integral gives area, while Green's Theorem gives volume?
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1answer
37 views

Calculate integral using Green's Theorem

Given the points $A(2,0), B(1,-1), C(1,0)$ and $D(0,-1)$ in $\mathbb{R}^2$, using Green's theorem I have to calculate the following integral: $$\int_{\Gamma}(x^4 -x^3e^x-y)dx+(x-y \arctan y)dy$$ ...
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1answer
64 views

On which regions can Green's theorem NOT be applied?

In my calculus book (Stewart), the theorem is proved for a simple region (I understand that this is being enclosed by a simple curve). But then it is specified that the theorem can be extended for a ...
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18 views

Integral on boundary of an open domain $D= D_1 \cup D_2$ with interface $\Gamma = \partial D_1 \cap \partial D_2$

A proof that I'm trying to understand involve the following \begin{align} \int_{\partial D} f(x)\cdot n(x) \phi(x)\ ds &= \int_{\partial D_1} f_{D_1}(x)\cdot n_{D_1}(x)\phi(x)\ ds + \int_{\...
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20 views

Evaluate the line integral $\int_C \vec{F}.\vec{dr}$ $\vec{F} =\langle2y, xz, x+y\rangle$

Evaluate the line integral $\int_C \vec{F}.\vec{dr}$ where $\vec{F} = \langle2y, xz, x+y\rangle$ and $C$ is the curve of intersection of the plane $z = y+2$ and $x^2 + y^2 = 1$ with orientation as ...
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18 views

Line integral bounded by vertices of a square

Let C be the square with four corners $$(1,1), (-1,1), (-1,-1), (1,-1)$$ oriented counter clockwise.How would I calculate $$\int_{C}x^3\vec{i}+(x^2y)\vec{j}ds$$ I have attached an image of my steps ...
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1answer
30 views

square $ [3,5] \times [1,4] $ theorem Green

I need to use a line integral for the square with vertices $(3,4), (5,4), (5,1), (3,1)$, using the green theorem i have $\displaystyle\int_3^5-\frac{15}{2}(4x^2+3)dx =-30\int_3^5 x^2dx-\frac{45}{2}\...
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2answers
56 views

Green theorem verifying problem

I am trying to solve a Green's theorem verify problem. Here is the problem: $Verify\;Greeen's\;theorem\;in\;the\;plane\;for \int_c\{(xy\;+y^2)dx\;+x^2dy\}, \;where\;C\;is\;the\;closed\;curve\;of\;the\...
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1answer
74 views

How to understand the singularity of Green's Theorem?

I'm learning complex analysis right now where I encountered the line integral of $\frac{1}{z}$ over a circle containing the origin, e.g. $\int_{\Gamma}{\frac{1}{z}}dz$, where $\Gamma: z(t) = e^{it}, t ...

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