Question

state and prove green theorem

Answer 1

$<b><u>Here Is Your Answer</u></b>$

To indicate that an integral $\ds\int_C$ is being done over a closed curve in the counter-clockwise direction, we usually write $\ds\oint_C$. We also use the notation $\partial D$ to mean the boundary of $D$ oriented in the counterclockwise direction. With this notation, $\ds\oint_C=\int_{\partial D}$.

We already know one case, not particularly interesting, in which this theorem is true: If $\bf F$ is conservative, we know that the integral $\ds\oint_C {\bf F}\cdot d{\bf r}=0$, because any integral of a conservative vector field around a closed curve is zero. We also know in this case that $\partial P/\partial y=\partial Q/\partial x$, so the double integral in the theorem is simply the integral of the zero function, namely, 0. So in the case that $\bf F$ is conservative, the theorem says simply that $0=0$.

$<i><u>Hope The Above Answer Helped.</u></i>$

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