Is it permissible for a female university student to cover her face with her hand to follow the professor's explanation on the board, knowing that she only sees the professor's hand, and is it permissible for her to look at the professor's shoe to determine her location and avoid looking at her, or is it better for her to wait until the lecture ends and obtain the summaries from her female classmates?

There is no harm in writing down the lesson while the teacher is writing it. If your gaze falls upon something impermissible, avert your gaze from it, for the Prophet, peace and blessings be upon him, said: "Do not follow a glance with another glance, for the first is yours but the last is not." The first glance referred to is an accidental glance, based on the hadith of Jarir, may Allah be pleased with him, who said: I asked the Messenger of Allah, peace and blessings be upon him, about an accidental glance, and he said: "Avert your gaze." As for looking at shoes or what is not considered 'awrah (private parts), there is no harm in it unless there is an element of fitna (temptation) in it. It is preferable to limit one's gaze to what is necessary only.