I remember very clearly, even though it was many years ago now, the preparation work that I did for my qualifying exams at MIT. I recently re-discovered some of the theory and equations that I summarized in own preparations. At that time about 1/3 of all Ph.D. students in the Mechanical Engineering department failed the qualifying exams, and I was determined that I was going to pass! The tests consisted of two days of both written and oral tests in four core areas. I had chosen: solid mechanics, fluid mechanics, thermodynamics, and dynamics.
I am very happy that I passed the exams.
Here is a free E-book that I created that summarizes some of the equations that I “collected” as part of that intense study. Hopefully someone will find the equations useful.
Best of luck with your own educational journey!