Things EVERY applied Mathematicians SHOULD know how to do or use (but probably don't).

Phase plane analysis

Non-dimensionalization

Reverse the axes: It is often the case that one wants to find (i.e. plot) a function u(x), where it is known that x=F(u(x)). The easiest way to plot u is to plot x as a function of u, and then reverse the axes.

Fredholm alternative (in my view the most important theorem in Applied Mathematics): Theorem: If L is a bounded linear operator on the Hilbert space L:H -> H with closed range, then the equation Lu=f has a solution if and only if < f,v>=0 for every v in the null space of L* (the adjoint operator of L).

fast-slow analysis (quasi-steady state analysis)

The resultant: The resultant is the determinant of the Sylvester matrix. The important property of the resultant is that R(f,g,x) is the product of all differences of all the roots of the polynomials f(x) and g(x), and therefore R(f,g,x)=0 if and only if f(x) and g(x) have simultaneous roots.