What Does 'Vanishes Identically' Mean in Mathematical Context?

[Ratzinger]

What does it mean?

the covariant derivative/ the torsion/ the eigenfunction...vanishes identically

When and where does what vanish identically?

 

[HallsofIvy]

"vanishes" means equal to 0. (I used to sit and wait for the "x" to vanish from the textbook. It never did!)

"Vanishes identically" means that the function is identically 0: 0 for all values of the variables.