# 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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### Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy$$ The Cauchy-Riemann equations have the ...
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### Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent ...
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### Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...
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### How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
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### Differential equation - Green's Theorem

I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ ...
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### Using Green's Theorem to Express the Integral $I=\int_C (Pdx+Qdy)$ as an expression of $I_i=\int _{C_i} (Pdx+Qdy)$

Let $p_1,...p_n$ be points in $\mathbb{R}^n$. Let $P(x,y), Q(x,y)$ be functions with continuous derivatives in $D=\mathbb{R}^2\setminus\{p_1,...p_n\}$ such that $Q_x-P_y=1$ for all $(x,y)\in D$. For ...
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### green's second identity application

I need to use the green's second identity in order to prove the following equality: $$\int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
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### Double Integral Formula for Holomorphic Function on the Unit Disc (Complex Plain)

While studying some complex analysis , I encountered the following problem: "Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$.Prove that for every $\zeta \in \mathbb{D}$ the ...
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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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