(1) Describe the level surfaces of $f(x,y,z) = sin(2x+y-z)$. For what values of 'c' do level surfaces exist?
For this one I set the function equal to c and tried to put it in a more manageable form. The result was $2x+y-z = arcsin(c).$
I think this is a plane, and I know the domain of arcsin is from -1 to 1. So is the answer simply "A plane with slope 2 in the x direction, 1 in the y direction, and 1 in the z direction. The level set exists when -1$\le$c$\le$1" ?
(2) Describe the level set of $F(x,y,x) = xyz$ which contains the point (0,2,-3)
When you set the expression $xyz$ equal to c and plug in (0,2,-3), you get the 0-level set obviously. But from there I don't really know how to find the answers to this question.
(1) Note that if $\sin(2x+y-z)=c$, we also have $\sin(2x+y-z +2\pi n) = c$ for every natural number $n$. So you get more than just one plane as a level set.
(2) If $xyz = 0$, what must be true about either $x$, $y$, or $z$?