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I wonder which expression can be used in my following example?

Our teacher was a really smart person. At the second session, he know everyone's name ................

  1. off the top of his head from the knowledge you have in your memory:

    • What's the capital of Mauritania?---I don't know off the top of my head, but I could go and look it up.

  2. by heart
    If you know something such as a poem by heart, you have learned it so well that you can remember it without having to read it.

    • Mack knew this passage by heart.
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    You might want to update the title to include the t from heart. (I would also say off by heart instead of just by heart, but I think they're synonyms so could just be my dialect)
    – Smock
    Aug 7, 2019 at 9:58
  • @Smock I'm pretty certain they are synonymous
    – Gamora
    Aug 7, 2019 at 10:03
  • You don't need to qualify that idiomatically. It's sufficient to say "he knew all the student's names", or you could qualify it like, "he remembered everyone's names" or "he could recall every student's names despite having only met us once", etc. As a native English speaker, "by heart" is probably more correct in this case than "top of my head", but I feel like "by heart" does not normally apply in this type of case.
    – David Zemens
    Aug 7, 2019 at 17:57

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Generally, off the top of my head is usually used to refer to the first thing that pops into your head.

What's the capital of Mauritania?---I don't know off the top of my head, but I could go and look it up.

This is a valid thing to say but you could also say:

I don't know off the top of my head, but it will come to me.

Meaning that it's not immediately available in your mind, but you do know the answer.

By heart implies something slightly different. To know something by heart it to know it inside out. You'd never hesitate to remember something you know by heart because you know it so well.

These are just my opinions and interpretations of the phrases (since you have already supplied the dictionary definitions).

However, given those slight nuances, I would lean towards using:

Our teacher was a really smart person. At the second session, he know everyone's name by heart.

...since your sentence seems to imply that he had picked them up really easily and knows them inside out, rather than just that that he had them in the forefront of his brain.

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A better (IMO) dictionary definition from Lexico is

off the top of one's head
PHRASE

Without careful thought or investigation.
I can't tell you off the top of my head

So in the case of the teacher remembering names, you say

Our teacher was a really smart person. At the second session, he knew everyone's name by heart.

The first phrase is used where you don't "know something by heart", but say the first thing that comes to your mind.

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