Author Topic: Help with attempting questions 17 -20 1.6  (Read 1657 times)

Nikita Dua

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Help with attempting questions 17 -20 1.6
« 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
« Last Edit: October 09, 2018, 05:09:14 PM by Victor Ivrii »

Arjaan

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Re: Help with attempting questions 17 -20 1.6
« Reply #1 on: October 14, 2018, 10:34:02 PM »
For question 20, you just need an f to satisfy that equation. In other words, you need an f such that dF/dx=f and dF/dy=if. This means that dF/dx=-i(dF/dy). This is a differential equation that can be satisfied by e^(y-ix+c)+c. I'm not 100% sure about it but it seems right. I got the same thing as you did for 17.

Victor Ivrii

17. Simply $Pdx+Qdy=dg$