Arithmancy. Hermione takes this class, but Harry does not so we never actually meet Professor Vector.
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Harry Potter and the Deathly Hallows Part 1 isn't set at Hogwarts and Professor McGonagall was still teaching there. She appears in the second half of the movie when Harry, Ron and Hermione go back to Hogwarts.
A music class only appears in the Harry Potter video games and the professor remains unnamed.In the movie Harry Potter and the Prizoner of Azkaban, the choir is conducted by Professor Flitwick.
Charity Burbage is a character from the Harry Potter books. She was the Muggle Studies professor at Hogwarts and appeared in Harry Potter and the Prisoner of Azkaban and Harry Potter and the Deathly Hallows. She is executed by Voldemort after defending muggle-borns in the Daily Prophet.
She is a teacher at Hogwarts. The strict but passionate leader of Gryffindor, she is also the deputy head of Hogwarts and the Transfiguration teacher.
Professor McGonagall is the transfiguration professor at Hogwarts and can turn into a cat. She doesn't follow Harry around though, the cat that does that is Filch the caretakers cat Mrs Norris, who is not a transfigured human.