I've been practising functions of several variables for college and I've been working with circles all the time $(x^2 + y^2)$, however, I still can't figure out how to solve non circular shapes, as far as I know by research $(x^2 – y^2) $ represent two queues that pass through the origin, and when you plot the function on 3D using the Google's plot you get this:
Google search for function's plot
The thing I don't understand is why does it look like a wave?, how do you determinate the Z/Height values for each queue and why X queue starts from below 0 and gains Z while Y starts from 0 and loses Z?
I would like that someone demonstrate to me (also including level curves) how to solve this exercise, sorry if it's an obvious question but I can't figure it out, really, plus, teachers don't help.. Thank you.
Best Answer
If you freeze $y$ (hence work in the vertical plane $y=Y$), you get curves $$z=x^2-Y^2$$ which are upward parabolas with their vertices at $(0,Y,-Y^2)$. The latter curve is obviously a downward parabola with its vertex at the origin.