MHB Locating Points for Vector Field $F$: $F_x=0$, $F_y=0$, and $|F_x|=1$

[paulmdrdo1]

Given Vector Field: $F=2(x+y)\sin\pi za_x-(x^2+y)a_y+\left(\frac{10}{x^2+y^2}\right)a_z$ specify the locus of all points at which a.) $F_x=0$ b.) $F_y=0$ c.) $|F_x|=1$

please help me get started with this. thanks!

 

[Sudharaka]

Given Vector Field: $F=2(x+y)\sin\pi za_x-(x^2+y)a_y+\left(\frac{10}{x^2+y^2}\right)a_z$ specify the locus of all points at which a.) $F_x=0$ b.) $F_y=0$ c.) $|F_x|=1$

please help me get started with this. thanks!

Hi paulmdrdo, :)

I am assuming that \(a_x,\,a_y\mbox{ and }a_z\) stands for basis vectors of some coordinate system. For (a), take the partial derivative of $F$ with respect to $x$. You will get,

\[2\sin(\pi z)a_x-2xa_y-\frac{20x}{(x^2+y^2)^2}a_z=0\]

\[\Rightarrow z\in \mathbb{Z} \mbox{ and }x = 0\]

For (b) notice that the coefficient of \(a_y\) is $-(x^2+y)$. Therefore the coefficient of $a_y$ in $F_y$ would be 1. Hence there aren't any points at which $F_y=0$.

For $(\mbox{c})$ you will get,

\[2\sqrt{\sin^{2}\pi z +x^2+\frac{100x^2}{(x^2+y^2)^4}}=1\]

This equation gives out all the points at which $|F_x|=1$. I am not sure whether we can simplify further. I hope somebody else might be able to come up with a better solution. :)