a potential big difference in the work environments of the these three.
the three being Mathematics, Physics, and EE (or engineering). I think there is a common misconception that focusing on one academic discipline necessarily limits one's career opportunities. In reality, that is not the case. One's career opportunities are limited by oneself.
When I was reaching the end of my physics studies (and switching to nuclear engineering), I followed what other physics undergrads were doing. What I discovered was that the majority of physics majors were going into related fields, as much as some were continuing on to grad school. One physics major ended up working for Halliburton doing well-logging, which is what a petroleum engineer, nuclear engineer, ME, or EE might have done.
Other physics majors went to work for companies like Exxon, Shell, or other oil companies, or energy companies, or technology conglomerates like GE, Westinghouse, or aerospace comanies like Boeing, Lockheed, McDonald Douglas, . . . or DOE, or NASA, i.e. government research organizations.
The young math major I hired went on to work for a financial services company. There are plenty of math majors working for Wall Street companies. They tend to focus on 'mathematical models' - no surprise there.
As for my colleagues in nuclear engineering - most stayed in nuclear - working for a utility, national lab (e.g. ORNL, ANL, INL/INEL, Sandia, Los Alamos, LLNL, . . ), or NASA. Several went outside nuclear into related areas such as safety and failure analysis. Others went on to work for engineering or technology companies, e.g. one went to work in the power division and markets turbines.
Companies like IBM and GE have corporate research labs which hire mathematicians, physics and engineers of all kinds.
The key is to be diversified in one's academic program. Keep the mind and eyes open - and don't limit oneself.
