What Does 'Vanishes Identically' Mean in Mathematical Context?

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"Vanishes identically" refers to a mathematical function that is equal to zero for all values of its variables. This concept is often discussed in the context of covariant derivatives, torsion, and eigenfunctions. The term emphasizes that the function does not just approach zero at certain points but is zero everywhere in its domain. Understanding this terminology is crucial for clarity in mathematical discussions. The phrase highlights the importance of distinguishing between functions that are zero at specific points versus those that are zero universally.
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What does it mean?

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

When and where does what vanish identically?
 
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"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.
 

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