The vector calculus formulation is not cumbersome and confusing, and we should not replace it. Many problems are simplest in vector calculus. Also, vector calculus formulation gives some insights that are not evident in other formulations. It's also extremely enlightening from a pedagogic perspective when you are learning physics. Of course if you're a mathematician and you don't want to learn physics, than vector calculus is just a distraction because it's idiosyncratic and doesn't connect with anything else.
The differential forms formulation, which in many other cases (but not all!) is more useful to work with than the vector calculus formulation should probably be replaced with geometric algebra though. They are basically the same thing, but differential forms are more idiosyncratic, and some symmetries are not apparent. Neither are useful as an introduction to the aspiring physicist or engineer though.
The differential forms formulation, which in many other cases (but not all!) is more useful to work with than the vector calculus formulation should probably be replaced with geometric algebra though. They are basically the same thing, but differential forms are more idiosyncratic, and some symmetries are not apparent. Neither are useful as an introduction to the aspiring physicist or engineer though.