TITLE: Why do they use $\equiv$ here?
QUESTION [7 upvotes]: I thought I had pretty much figured out the difference between $\equiv$ and $=$. Then I came across this while reading about partial derivatives (in Colley's Vector Calculus):
$$
\frac{\partial^2f}{\partial z^2} = \frac{\partial}{\partial z} \left(\frac{\partial f}{\partial z}\right) = \frac{\partial}{\partial z} (y^2) \equiv 0
$$
when $f(x,y,z)=x^2y+y^2z$. Why do they use $\equiv$ in stead of $=$ here?

REPLY [15 votes]: $\equiv$ is often used (between functions) to mean they are identical (instead of being equal at some point).  in your example this means identically $0$.  if someone write something like $f(z)=g(z)$ it might be thought these functions are equal at some point $z$ instead of every point $z$, so you could write $f\equiv g$ or $f(z)\equiv g(z)$ to mean they are equal everywhere.