Applied Mathematics consists of two words: 'applied' and 'mathematics'. In general, Applied Mathematics also include two areas. One area is the Mathematics related to Application, which may be called APPLICABLE MATHEMATICS. This a subset of traditional mathematics. The other area is the APPLICATION OF MATHEMATICS, i.e., the study and solution of scientific and engineering problems using mainly mathematics as the tool. This is not a subset of the traditional mathematics and is beyond traditional mathematics.

Process of Applied Mathematics

These two areas of applied mathematics are related through the Process of Applied Mathematics, which involves 5 steps:

1.Observations; Experimentations; Collection and Organization of Information.

2.Establishment of Mathematical Model.

3.Create New Mathematics and Mathematical Tools; Extension and Development of Known Mathematics.

4.Solving the Mathematical Problem.

5.Check with the Experiments and Observations.

The best model applied mathematician is Newton. In order to understand the motion of heavenly bodies, he took the following 5 steps:

1.Organnization of existing observation data and personal astronomical observations.

2.Laws of mechanics and law of gravitation.

3.Invention of calculus.

4.Solving the problem of motion of heavenly bodies.

5.Agreement with observation.

Newton was a rare genius. Scientific development in Newton's time was relatively simple in scope. Hence Newton could single-handedly complete an entire Process of Applied Mathematics. Most of us cannot compare with Newton. The diversity and complexity of modern scientific development force us to be specialists in a narrow discipline. Even for problems of rather small scope, few persons can complete a Process of Applied Mathematics all by himself. Thus we shall consider those scientists who undertake some part of the work in the Process of Applied Mathematics, and who are keenly aware of the underlying Process, "applied mathematicians".

The APPLICABLE MATHEMATICS are concerned with the steps 3 and 4 in the Process. Nowadays, most so-called applied mathematicians in the US, and almost all those in China are doing applicable mathematics. Unfortunately, most of them do not have the Process of Applied Mathematics in mind. Thus they miss the essence of the problem and dwell too much on unimportant mathematical technicalities. That is why Hardy would term it trivial mathematics and earned it the scorn from pure mathematicians.