I have an integral surface $z = z(x, y)$.

Writing this integral surface in implicit form, we get

$$F(x, y, z) = z(x, y) - z = 0$$

I am then told that the gradient vector $\nabla F = (z\_x, z\_y, -1)$ is normal to the integral surface $F(x, y, z) = 0$.

First of all, how was this calculated? I understand how the gradient is calculated, but I don't understand how it was calculated in this case?

And lastly, where did the $-1$ come from and why? Couldn't they also have had $\nabla F = (z\_x, z\_y, 1)$, where this would just be the normal vector in the other direction? Why and how did they pick the $-1$ direction instead?

I apologise. My vector calculus understanding is not particularly strong, and I strive to improve it.

Thank you for any help.