The hyper surface in my lectures is defined as a single function of coordinates which is a constant-
$f(x)= c$
and it’s stated that the normal to the hyper surface is given as
$x_\mu= \nabla_\mu f(x)$
But isn’t the above expression just 0 since $f(x)= constant$ implies
$\nabla_\mu f(x)= \nabla_\mu (constant) =0$
Won’t the above expression be just always 0?
It’s not that $f$ is a constant function, like $f(x, y, z) := 7$. Rather a hypersurface is the set of points where $f$ attains one particular value, like $\{(x,y,z) \in \mathbb R^3: f(x,y,z) = 7\}$. In particular, $f$ is changing in every direction not parallel to the hypersurface so it’s far from a constant.
Edit: Consider $f(x, y, z) = x^2 + y^2 + z^2$. The set $\{(x, y, z) \in \mathbb R^3: f(x, y, z) = 1 \}$ is a sphere with radius 1. The gradient of $f$ is $\nabla f = \langle 2x, 2y, 2z \rangle$, so you see that partial derivatives are nonzero functions. Moreover, for $(a, b, c)$ on the sphere, $(\nabla f)(a, b, c) = \langle 2a, 2b, 2c \rangle$ is a normal vector to the sphere.