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« **on:** October 09, 2018, 05:01:52 PM »
I am not sure on how to approach the questions 17-20 from 1.6.

For 17 I started off using the green's theorem

$$ \int_{\gamma} (Pdx + Qdy) = \iint_{\omega} \Bigl[\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \Bigr]dxdy

$$

Since $Pdx + Qdy$ is exact differential $P = \frac{\partial g}{\partial x}$ and $Q = \frac{\partial g}{\partial y}$

$$\frac{\partial P}{\partial y} = \frac{\partial ^2 g}{\partial x \partial y} \\

\frac{\partial Q}{\partial x} = \frac{\partial ^2 g}{\partial x \partial y}$$

So $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0$

Hence $$\int_{\gamma} (Pdx + Qdy) = \iint_{\omega} \Bigl[\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\Bigr] dxdy= 0$$

Not sure if this is correct and how to proceed with 18-20