APMA Fundamentals

The cybernetic revolution that began on the eve of twenty-first century effected our life in all dimensions. The collection, organization, transformation, and interpretation of information are cornerstones of science, industry, business, and government. Each aspect of information management uses mathematics as a tool.

Applied Mathematics (APMA for short) has undergone a real metamorphosis during this century, partly due to the birth and development of computers. It is so active that entire periodical magazines are now devoted to it, and one single tutorial can only reflect part of its vitality. Here is an attempt to face this challenge in a concise-although rigorous-manner. Applied Mathematics is a general and powerful language. This tutorial is based on utilization of powerful software packages and focuses on applications, justifying the elaboration of such an abstract language.

Applied Mathematics is employed in most of computational physics and information technologies. The tutorial includes the following topics:

• Complex Variables
• Series and Limits
• Differentiation
• Integration
• Vector Calculus
• Tensors
• Calculus of Variation
• Classical Mechanics
• Quantum Mechanics

There are many computing resources available that will aid in your exploration and understanding of differential equations. The main software packages that we recommend are

• Matlab® (Octave is a free software similar to matlab)
• Maple™ (CAS)
• Mathematica® (Mathics is a free software similar to Mathematica)
• R (free software)
• Maxima (free computer algebra system)
• Sage (free computer algebra system)
• SymPy (free computer algebra system)
• Python (free software)
• Julia (free software)

All these packages have the syntax commands closely related to traditional pen-and-paper mathematical language. According to the National Council of Teachers of Mathematics, the goal of teaching mathematics is to help all students develop mathematical concepts including the abilities of students in converting symbolic information into graphical and technology. This can be achieved and constructed using programming that is useful at any stage of mathematical thinking especially in conjecturing. In our days, computational thinking should be considered a fundamental analytic skill in education, along with reading, writing, and arithmetic. This is a vision for the twenty-first century supported by both the National Research Council of the Academy of Science and the National Council of Teachers of Mathematics.

The author feels deeply that computing should be an integral part of any course. Graphics capabilities alone warrant its use. Of course. technology is not a substitute for actual knowledge of material. Technology is a tool to gain insights into complex problems and it can be a valuable tool to better understand difficult concepts and learn the scientific methods of inquiry. The author believes that computing brings additional insight and helps to develop curiosity that theory alone cannot achieve. As a result, the qualitative aspects of linear algebra could be explored in greater depth.