The function $f:\mathbb R^{3}\times \mathbb R \rightarrow \mathbb R$ is defined by the following formula:
$f(x,t)=W(x \cdot v - ct) $
where $W: \mathbb R \rightarrow \mathbb R$ is a given function, $v \in S^{2}\subset \mathbb R $ is a unit vector, and $c>0$ is a scalar.
We need to:
- determine $\nabla f$, and
- determine $f^{-1}(y)$ for an arbitrary $y \in \mathbb R$ in the range of $f$ (in order to describe the level sets.
My attempts for both questions are all over the place. I understand that $\nabla f$ is a vector of partial derivatives, and the steps taken to derive the inverse of a function. However, I don't completely understand how to apply these properties to this particular function.
I would really appreciate an explanation or some guidance.