What is the ruling on swearing unintentionally with "that I will not teach at my other friend's place," and is it specific to the time of the first teaching session or is it general, and does an expiation become incumbent upon me?

If your intention when swearing the oath was not to absolutely prevent your friend from explaining in the presence of the other one, then you have not broken your oath and no expiation is due from you if she explained to you in another place or at another time, and she did not behave in the manner that displeased you. This depends on the context of the oath and the reason that prompted it. If there was no specific intention, then one refers back to the reason for the oath and what provoked it.