Hand waving is the most integral part of calculus. The idea is to pretend that arguments with holes are valid and rigorous.

Techniques

There are many techniques for hand waving calculus. The first is loud speech. Loud speech allows you to say things without anybody talking back, for fear they will get yelled at. The second is triviality arguments. In this, you say, "Proof is trivial". Usually people will accept you without looking at this themselves. A closely related method is obviousity arguments. In support of some step, you say it is "obvious" or "clear", etc. An expanded version of this method is "The proof is left to the reader (or student.)" As the reader or student has more important priorities, no one will care to check things out. Another method is to make references to obscure, not necessarily related theorems, and connect the start and end of your argument to these references. This will give the illusion of causal contiguity.

Uses

As you know, mathematics cannot proceed without rigor. This makes things slow. With hand waving calculus, we can proceed much faster and in fact can prove many things that it was impossible to prove before. The key concept is that there is an illusion of rigor while it is not really present, so people accept it.

See also