# 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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### Green's theorem for piecewise smooth curves

Green's theorem is usually stated as follows: Let $U \subseteq \mathbb{R}^2$ be an open bounded set. Suppose its boundary $\partial U$ is the range of a closed, simple, piecewise $C^1$, positively ...
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### Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$

Given $f: \mathbb{R^3} \rightarrow \mathbb{R}\in C^2$ s.t. $∆f > 0$ prove that $\frac{d\int_{\partial B(0,r)}f}{dr} > 0$. I know that this is just using Green's first formula, but I'm having a ...
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### Evaluate the contour integral (Most likely without Green's Theorem)

$\int_{c}\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ where $C$ is the triangle with vertices at $(5,5), (-5,5),$ and $(0,-5)$ traversed counterclockwise. (Hint: Be careful about the hypotheses of any ...
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### Green's Theorem with change of variables

Evaluate $$\int_C F dr$$ $$F =< x2, xy >$$ $$C: x^2/4^2 + y^2/9 = 1$$ With y ≥ 0 positively oriented. For the circle $$u^2 + v^2 = 1$$ $$u=x/a$$ $$v=y/b$$ $$x=au$$ $$y=bv$$ ...
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### (Closed) Line integral of Conservative Field.

Suppose we have a conservative Field $\vec F: D' \subseteq R^2 \rightarrow R^2$ where D is a set of points inside a closed curve (for example all the points inside a circle). Say we have subset of D',...
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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 ...