Expert: Abe Mantell - 9/2/2008

Question
Prof;

I need to use polar coordinates to describe the level curves of the function defined by f(x,y) = 2xy/(x^2+y^2)

All I know is that in polar coordinates the function f(x,y) becomes P(r,θ) = sin(2θ), but I have no idea what is meant by finding level curves using polar coordinates, why are they an adventage in this case? how do I do it???

I'm stumped and any help would be greatly appreciated.

Thank You.

Answer
Hello John,

Polar form often helps one sketch and understand a complicated function.
It is not all that obvious what the level curves of f(x,y) look like in
rectangular form, but in Polar form...
x=r*cos(theta), y=r*sin(theta) ==> x^2 + y^2 = r^2, thus,
f(r,theta)=2*r*cos(theta)*r*sin(theta)/r^2 = 2cos(theta)sin(theta),
which simplifies, using a double angle formula, f(r,theta)=sin(2*theta)

OK?

Abe