Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started by: Nikita Dua on October 09, 2018, 05:01:52 PM

I am not sure on how to approach the questions 1720 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 1820

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^(yix+c)+c. I'm not 100% sure about it but it seems right. I got the same thing as you did for 17.

17. Simply $Pdx+Qdy=dg$
18. Use Green's (not green's)
and so on
Arjaanunreadable