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MAT334-2018F => MAT334--Lectures & Home Assignments => Topic started by: Nikita Dua on October 09, 2018, 05:01:52 PM

Title: Help with attempting questions 17 -20 1.6
Post by: Nikita Dua 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
Title: Re: Help with attempting questions 17 -20 1.6
Post by: Arjaan 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.
Title: Re: Help with attempting questions 17 -20 1.6
Post by: Victor Ivrii on October 16, 2018, 06:53:07 AM
17. Simply $Pdx+Qdy=dg$
18. Use Green's (not green's)
and so on

Arjaan--unreadable