"Know about" vs. "know of"

Solution 1:

I would use the expression "I know Smith" if I have had personal experiences with the person.

For me, "know about" means that I have heard many things about this person, or have researched him (such as if I were to say "I know about Einstein, he invented general relativity"), while "know of" suggests that I have merely heard of this person in passing, or have a very limited knowledge of this person (such as if I were to say "I know of Madame Curie", the emphasis being on the word "of", and usually as a reply to someone asking "Have you heard of Madame Curie?").

Therefore, I would say your friend is wrong in his characterization of the former, and correct in his characterization of the latter.

Solution 2:

My immediate thought was the opposite of your friend:

I know of tensor calculus (I have heard the name, but that is about all I know.)

I know about tensor calculus (I have learned this subject and can do the calculations.)

I know all about tensor calculus (I am an expert)