What’s the difference between “$≡$” and “$:=$”

Question

When I write:
$$d\zeta \equiv \omega dt$$
And:
$$d\zeta := \omega dt$$
Is there any difference between the two expressions ? Are they both correct ? What's in general the difference between $\equiv$ and $:=$ ? Seems to me that both of them mean "is defined by", even though I saw $\equiv$ more often with functions and $:=$ with variables.

Answer 1

$\equiv$ is a symbol for “identically equal to”. For example:

$$\tan x\equiv\frac{\sin x}{\cos x}$$

means “the left side is always equal to the right side, no matter what the variables (in this case $x$) are”. This is a fairly old-fashioned symbol, and these days you would more often see the same thing written as:

$$\tan x=\frac{\sin x}{\cos x}\text{, for all }x\in X$$

where $X$ is explicitly given (say $X=\mathbb R\setminus\{k\pi+\pi/2\mid k\in\mathbb Z\}$), which conveys a bit more information anyways.

The $:=$ symbol means “is defined as”, and is used in definitions, to introduce a new symbol or a new notion. For example, you can use it to define the tangent of an angle:

$$\tan x:=\frac{\sin x}{\cos x}$$

This use is different from the previous one: in the previous use of $\equiv$ (or $=$) we merely claimed that the tangent of $x$ always has the same value as sine divided by cosine. Here we introduce a previously (in a given context) unknown notion of a tangent, which becomes the same as sine divided by cosine by definition.