Gradient vector at some point Suppose you are given some function $f(x, y, z) = xy^3z^2$ and you are told to find its the gradient vector normal to the surface at some point (-1, -1, 2).
Is it right think that the gradient vector is normal to the surface as shown in the figure?

Also, if you substitute the point you get a gradient vector at that point say $P$ ?

Because you have a function $f(x,y,z)$ of three variables, I assume you're talking about the level surface at a certain point, where given a constant $k$ the function has the value
$$f(x,y,z)=k$$
Then in that case the gradient vector
$$\nabla f(x,y,z) =\frac{\partial f}{\partial x}\hat{\imath}+\frac{\partial f}{\partial y}\hat{\jmath}+\frac{\partial f}{\partial z}\hat{k}$$
 is always normal to the level surface. Substituting any point $P$ will give you another vector normal to the level surface at that point $P$, but it may not be the same level surface as the original one.
