It was fascinating to learn how to solve certain classes of differential equations in one class, and then go to this class, set up some differential equations using conservation laws and common sense, and using what I just learned in the other class to solve them! Yes, from the perspective of the differential equations we set up. At this point, you are easily mixing tools from 4–5 different upper division courses.
But there is something funky going on when you analyze the solutions more in depth. This subject is much more fun and fascinating than just learning numerical methods and grinding through monotonous calculations. You are given tools that can be applied to countless topics, and as a consequence you get glimpses into many different disciplines.
We then return to the solution of problems in classical dynamics through the use of Lagrange's equations and Hamilton's equations, as well as Hamilton-Jacobi theory which is fundamental to quantum theory discussed towards the end of the book.
Although the subjects of Fourier series, Fourier and Laplace transforms, and integral equations, are not strictly applied mathematics, they are essential for the study of wave motions, including vibrating strings, sound waves and water waves, and for the study of heat conduction.
A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problem solving.
How To Solve Applied Mathematics Problems
It is suggested that effective instructional techniques for teaching applied mathematical problem solving resembles “mathematical laboratory” activities, done in small group problem solving settings.
One can try to formulate basic principles for solving problems in applied mathematics, and to a certain extent this can be done, but the best way is to study the solutions of a large representative selection of problems.
It is important to realize that solving problems in applied mathematics is strongly dependent on understanding, and is not just a matter of memory, although a knowledge of the relevant formulae plays a significant role.
All the problems can be solved in closed analytical forms in terms of elementary functions or simple integrals.
The book starts with an introductory chapter in which there is a survey of the range of subjects which are covered in the subsequent chapters by means of a small representative selection of problems, together with some general principles for solving problems in applied mathematics.