00:01
We're given the function f of xy is equal to x over the square root of x squared plus y squared when we are excluding the origin.
00:10
We first want to rewrite this in polar coordinates and then use this to help us find our level curves.
00:18
So first, recall that to convert the polar coordinates, we're going to let x equal to r cosine theta.
00:26
And we could let y equal to r sine theta, but we also have that r is equal to the square root of x squared plus y squared, and we already have that in the denominator, so that would be a more direct substitution.
00:42
So we have r, cosine, theta, all over r, so this is f of r theta, so the rs cancel out with each other, and then we have cosine theta.
00:54
So we're going to go ahead and replace f of r theta, with c.
01:00
And now we know that cosine's range is negative 1 to 1.
01:10
So those are the only possible level curves we can have for c.
01:14
And that would tell us if we'd get, so theta is going to be arc cosine or cosine inverse of c.
01:29
Or we can think about it instead as it'd be plus or minus cosine of the absolute value of c.
01:42
And this would only be when the absolute value of c is less than or equal to one.
01:50
Actually, strictly less than one.
01:53
So now, and also greater than zero...