I often wondered why a B.Sc. in physics has learned so much more about mathematics than what's in an undergraduate mathematics program. Is mathematics simple, easy, even relaxing compared to physics? It certainly seems that way. Well, seemed, for me anyway. I think I understand it now.
Undergraduate physics books are almost self containing with regards to mathematics. Mathematics is the language of physics enabling them to solve all sorts of problems. ( I am not surprised that innovations in physics were carried by the introduction of new mathematics to the field. Since they aren't trained in developing new mathematics they are also constrained by it. ) Large quantities of math are pumped into their brains without proof.
This video illustrates my point. It is lecture 3 from the MIT OpenCourseWare Physics I course.
This lecture is about vectors and how to add, subtract, decompose and multiply vectors. Decomposing vectors in 2 (or 3) dimensions is a key concept that will be used throughout the course.