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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How to evaluate integral of (x - y)(dx + dy) with Green's Theorem?

I want to evaluate the integral $\int(x - y)(dx + dy)$ along curve C where C is the semicircular part of $x^2 + y^2 = 4$ above $y = x$ from $(-\sqrt2, -\sqrt2)$ to $(\sqrt2, \sqrt2)$ using Green's ...
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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 ...
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find the simple closed curve of $F(x,y) = (y^3-6y)i + (6x-x^3)j$ using Green's Theorem which will have the largest positive value

$F(x,y) = (y^3-6y)i + (6x-x^3)j$ a. Using Green's Theorem, find the simple closed curve C for which the integral $∳F \cdot dr$ (with positive orientation) will have the largest positive value. b....
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An intuitive explanation for Green theorem and Divergence theorem

As my vector calculus exam is getting closer, I'm looking for intuitive ways to think about the different theorems we have to memorize. I think I have found a pretty intuitive way to think about the ...
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Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$

Calculate $\int_{C} \frac{x}{x^2+y^2} dx + \frac{y}{x^2+y^2} dy~$ where $C$ is straight line segment connecting $(1,1)$ to $(2,2)$ my question is , after calculating the integral using green theorem ...
Finding the maximum value of the integral $\int_{C}x^2y-2y^2-5y{dx} +(2xy-y^2x){dy}$
Find the maximum value of $\int_{C}x^2y-2y^2-5y{dx} +(2xy-y^2x){dy}$ , where C is closed curve with no self crossing taking in the positive direction. it is obvious that i need to calculate using ...