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

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $D=\{(x,y) \in \mathbb{R^2}: r^2 < x^2 + y^2 < R^2\}$ $F$ ...
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 $y\in \Omega$: ...
### 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 ...