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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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Line integral in proof of Green's theorem

In wikipedia page about Green's theorem the following equality appears: $$\int_{C_1} L(x,y)\, dx = \int_a^b L(x,g_1(x))\, dx$$ I do not understand it. Wikipedia page about line integral defines ...
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Show that $F_x(x,y) = P(x,y), \ F_y(x,y) = Q(x,y)$.

In Complex Variables and Applications by Brown and Churchill it comes: When the point $(x_0,y_0)$ is kept fixed and $(x , y)$ is allowed to vary throughout a simply connected domain $D$, the ...
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green theorem statement's meaning

Vector Calculus sector of 6.2 17. D is always on the left as we travel along C (C is the path of D) What is the meaning of the above statement? I can't understand ;(
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Area of a simple closed curve

Let C be a simple closed curve in a region where Green's Theorem holds. Show that the area of the region is: \begin{equation} A=\int_{C}xdy=-\int_{C}ydx \end{equation} Green's theorem for area ...
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Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field

Calculate the line integral on the negatively oriented unit circumference (like the hands of the clock) under the $F(x,y)=(e^x+x^2y,e^y-xy^2)$ field I have thought to do the following to solve this: ...
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Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ along a path

Find the work done by the force field $F(x,y)=(x^2+xy)\bar{i}+(xy^2)\bar{j}$ when a particle moves from the origin, along the $x$ axis to the $(1,0)$, then on the line segment that joins the $(1,0)$ ...
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Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined

Calculate $\int_CPdx+Qdy$ where $P(x,y)=xe^{-y^2}$ and $Q(x,y)=-x^2ye^{-y^2}+\frac{1}{x^2+y^2+1}$, $C$ is the boundary of the square determined by the inequalities $-a\leq x\leq a$, $-a\leq y\leq a$, ...
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Exercises of aplication of Green's theorem

Use Green's Theorem to find the limited area above the $x$-axis and below the circle of centers $C_1(0,1)$ and $C_2(2,1)$, both of radio equal to $1$. $C_1: x^2+(y-1)^2=1$ and $C_2: (x-2)^2+(y-1)^2=1$...
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Is this equality true? (cauchy integral + green's theorem)

Is this just TRUE? $$\oint_{\partial S} f(z) \, dz = i\iint_S \bigg[\frac{\partial{f}}{\partial{x}}+i \frac{\partial{f}}{\partial{y}}\bigg] \, dx \, dy$$ Because I'm using it without a proof, and I ...
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Proof of Green's theorem under assumption of only type 1 curve.

The proof of Green's theorem on Stewart's calculus book has assumption of the closed curve being both type 1 and type 2. Can one prove the theorem under assumption of being only type 1? Are there ...
let $F=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ and $R(t)=(\cos t,\sin t)$ (the curve is a circle with radius 1) now: \begin{equation} \int_{R}F_1.dx+F_2.dy = \int_{0}^{2\pi}-\sin t\ dt + \...