# 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 $C$.

408 questions
Filter by
Sorted by
Tagged with
20 views

### $\int_{C}^{}v dS$, 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 ...
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 $-\int_Uu\Delta v dx$ ...
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 $(1,0)$...
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 $G(\textbf{r},\textbf{r}')=\frac{1}{4\pi|\textbf{r}-\textbf{r}'|}$. ...
81 views

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 ...
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 ...
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 ...
43 views

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 ...
57 views

### 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'...
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 ...
62 views

57 views

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 $\displaystyle\int_3^5-\frac{15}{2}(4x^2+3)dx =-30\int_3^5 x^2dx-\frac{45}{2}\... 2answers 56 views ### Green theorem verifying problem I am trying to solve a Green's theorem verify problem. Here is the problem:$Verify\;Greeen's\;theorem\;in\;the\;plane\;for \int_c\{(xy\;+y^2)dx\;+x^2dy\}, \;where\;C\;is\;the\;closed\;curve\;of\;the\...
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 ...