# 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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### The work done by the force $\vec{F}$ on a particle

Question The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first ...
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### Could I use Green's Theorem here?

I want to solve for the line integral: $$\tag{1}\oint \alpha\nabla \phi_i\cdot \hat{\textbf{n}} ds$$ on the square boundary: $(0\le x \le 1, 0), (1,0 \le y \le 1), (1 \le x \le 0, 1),(0,1\le y \le 0)$ ...
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### Confusion over why Green's Theorem is usable in one situation with closed circle and not the other.

As I understand it, if a region contains the origin, Green's Theorem cannot be applied. However, one question in my issued lecture notes appears to contradict this and another one appears to follow ...
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### Caculate the line integral of a ellipse by a small circle.

Calculate the line integral $$\int_\gamma \frac{y\,dx+(1-x)\,dy}{(x-1)^2+y^2}$$ where $\gamma$ is the ellipse $x^2 + 4y^2 = 4$ traversed two laps in positive direction. So I have been given a ...
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### Stokes' Theorem and perimeter

As a consequence of Stokes' Theorem it seems that the perimeter of a closed curve $C$ can be obtained by choosing $F$ to be the vector field formed by rewriting the unit tangent $T$ of $C$ as a ...
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### Green's Theorem - Applying a translation to the vector field to make the integral easier to evaluate

I have to evaluate a line integral using Green's theorem: $$\overrightarrow{F}(x,y)=\langle y-\cos{y}, x\sin{y} \rangle$$ The curve $C$ is the circle $(x-3)^2+(y+4)^2=4$, oriented clockwise. It would ...
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### Green's Theorem circle in a circle (hole) when both are traversed in the same direction

Im struggling to understand how to apply Green's theorem in the case where you have a hole in a region which is traversed in the same direction as the exterior. For a workable example I want to ...
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### inferring sign of a double integral over a general region D from double integrals of all boxes that lie inside D and containing D

Suppose a double integral $$I_{D}:=\iint\limits_D f(x,y) dx dy$$ over a specific type of region $D=\{(x,y) : x_{\min}\leq x \leq x_{\max}, h_1(x) \leq y \leq h_2(x)\}$. Note that $h_1, h_2$ are ...
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### Calculating line integral on a vector field, help me find the mistake

Alright, so, I have vector field: $F=[p(x,y), q(x,y)]=[y^3+e^{x^2}, x^3+{\tan}^2y]$. I need to calculate $\oint_Lpdx+qdy$, where $L: x^2+y^2+4y=0$. I transform it to $x^2 + (y+2)^2 = 4$, i.e. a circle ...
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