Applied Mathematics

The primary objective of the course presented here is orientation for those interested in applying mathematics, but the course should also be of value or in using math­ to those interested in mathematical research and teaching ematics in some other professional context. The course should be suitable for college seniors and graduate students, as well as for college juniors who have had mathematics beyond the basic calculus sequence. Maturity is more significant than any formal prerequisite. The presentation involves a number of topics that are significant for applied mathematics but that normally do not appear in the curriculum or are depicted from an entirely different point of view. These topics include engineering simulations, the experience patterns of the exact sciences, the conceptual nature of pure mathematics and its relation to applied mathe­ matics, the historical development of mathematics, the associated conceptual aspects of the exact sciences, and the metaphysical implications of mathe­ matical scientific theories. We will associate topics in mathematics with areas of application. This presentation corresponds to a certain logical structure. But there is an enormous wealth of intellectual development available, and this permits considerable flexibility for the instructor in curricula and emphasis. The prime objective is to encourage the student to contact and utilize this rich heritage. Thus, the student's activity is critical, and it is also critical that this activity be precisely formulated and communicated.

1. Introduction.- 1.1. Vocational Aspects.- Applied mathematics is the vocational use of mathematics other than in teaching or mathematical research..- 1.2. Intellectual Attitudes.- In a technical effort, understanding cannot be disjointed into pieces corresponding to the academic disciplines. Technical understanding has a basically algorithmic character..- 1.3. Opportunities in Applied Mathematics.- Many of the possibilities for applied mathematics occur as part of research and development programs of the Federal Government. Technology advances may also open opportunities for applied mathematics in industry..- 1.4. Course Objectives.- The exercises and the student projects are an essential part of the course..- Exercises.- 2. Simulations.- 2.1. Organized Efforts.- Applied mathematics is usually part of a large effort under contract with the Federal Government and based on scientific and technical understanding. It is a team effort and documentation is essential..- 2.2. Staging.- The efficient use of resources requires that such efforts proceed in stages, each of which provides a decision basis for the next..- 2.3. Simulations.- Technical simulations permit decisions to be based on the scientific and technical understanding of the original situation..- 2.4. Influence Block Diagram and Math Model.- The basic understanding is expressed in the influence block diagram and the math model..- 2.5. Temporal Patterns.- The block diagram and the math model are supplemented by the flow chart, which describes the relations in time of the original situation. Specific scenarios are also used..- 2.6. Operational Flight Trainer.- The notions of influence block diagram and math model are illustrated in this example..- 2.7. Block Diagrams.- Block diagrams originally referred to equipment. In analog computers these became associated with the math model..- 2.8. Equipment.- The equipment includes the computer and the input and output devices required for the simulation. The objectives of the simulation determine the requirements..- 2.9. The Time Pattern of the Simulation.- The basic time pattern of the simulation is based either on an advance by fixed time intervals or by critical events. Provision must be made for input and output..- 2.10. Programming.- The structure of the program should be modular and subject to an executive program. The numerical procedures must be determined with the required accuracy, stability, and range..- 2.11. Management Considerations.- The total effort in the simulations must be scheduled to permit the efficient use of resources such as manpower and facilities..- 2.12. Validity.- The mathematical formulation of understanding can best be understood in terms of its historical development..- Exercises.- References.- 3. Understanding and Mathematics.- 3.1. Experience and Understanding.- Understanding permits us to cope with an environment by using past experience patterns in a mental exploration of possibilities..- 3.2. Unit Experience.- A flow diagram for a unit experience indicates the adjustment between understanding and the interaction with the environment. The validity of knowledge is associated with this adjustment..- 3.3. The Exact Sciences.- For situations in their milieus, the exact sciences produce a block diagram analysis whose blocks correspond to concepts based on patterns of experience and whose math model yields prediction and control..- 3.4. Scientific Understanding.- Scientific understanding is effective because it represents a long-range adjustment of concepts and math model to match experience. But this adjustment involves complications that must be understood..- 3.5. Logic and Arithmetic.- Many aspects of experience can be usefully formulated in terms of the concepts associated with finite sets and the natural numbers..- 3.6. Algebra.- Algebra represents an abstraction of the properties of numbers that greatly supplements the logical possibilities for elementary arithmetic..- 3.7. Axiomatic Developments.- When notions have an approximate symbolic representation it is possible to set up an axiomatic development starting with “axioms” and using agreed upon logical principles. One has such a development for the natural numbers, abstract algebras, and geometry..- 3.8. Analysis.- The development of the exact sciences required a further expansion of mathematics called analysis. In modern times analysis has been axiomatically structured on the basis of set theory. This has led to many mathematical developments..- 3.9. Modern Formal Logic.- There has been an effort to formulate a symbolic system representing “true logic” independent of conceptual experience. However, there is a widespread opinion that applied mathematics does not need to conform to the intellectual restraints represented by this point of view..- 3.10. Pure and Applied Mathematics.- The requirement of a purely logical development of mathematics eliminates all but one type of conceptual experience and produces a significant isolation for pure mathematics. To counteract this the applied mathematician must understand the historical development of the exact sciences and mathematics..- 3.11. Vocational Aspects.- The applied mathematician must assume responsibility for incorporating the effective use of the math model and the computations into the simulation..- Exercises.- References.- 4. Ancient Mathematics.- 4.1. Ancient Arithmetic.- Arithmetic is an essential base of civilized cultures. Egyptian arithmetic was based on a decimal notation for integers with a binary procedure for multiplication and division..- 4.2. Egyptian Mathematics.- The Egyptians handled many problems by a rational arithmetic using mixed numbers in a special form. The notion of algorithm is clear..- 4.3. Babylonian Mathematics.- Babylonian mathematics used a sexagesimal notation, a system of tables using square and cube roots, and a “geometric algebra” equivalent to our present quadratic equations..- 4.4. Greece.- Higher education appeared in classical Greece. Euclidean geometry was considered an essential basis..- 4.5. Euclid’s Elements.- Geometry was a logical development from “first principles,” which were given in three forms―postulates, axioms, and definitions. The definitions are explanatory and can be divided into categories, each associated with a combination of concepts. The nature of generality in geometry and the formats of Euclid’s proofs are discussed in Proclus..- 4.6. Magnitudes.- The logical development of magnitudes was an essential aspect of geometry. Problems could also be formulated in “geometric algebra.” The Dedekind cut definition of real number is clearly associated with the notion of ratio in Euclid. Basic number theory is given in Euclid’s Book VIII..- 4.7. Geometry and Philosophy.- Geometry had a profound influence upon philosophy and shaped the ideas of developments from first principles and of “truth.”.- 4.8. The Conic Sections.- The conic sections are the plane sections of right circular cones. They satisfy geometric relations called “symptoms” equivalent to the modern equation of a locus. Appolonius generalized the notion of symptom. This permitted a classical procedure for obtaining tangents..- 4.9. Parabolic Areas.- Archimedes developed heuristic procedures for finding certain areas, and these procedures suggested the integral calculus. The resulting relationship was proved rigorously in two ways..- Exercises.- References.- 5. Transition and Developments.- 5.1. Algebra.- Diophantus considered problems involving numbers as such in a prose equivalent to our present algebraic manipulation of equations. Renaissance mathematicians developed modern elementary algebra, including the solution of the cubic and quartic equations. This algebra was an important element in the development of the calculus..- 5.2. Non-Euclidean Geometry.- Postulates 4 and 5 of Euclid yield a precise analysis of the geometry of the Euclidean plane. Variations of the fifth postulate yield the non-Euclidean geometries..- 5.3. Geometric Developments.- European interest in geometry led to analytic geometry and an expansion of synthetic geometry. Modern forms of synthetic geometry are based on set-theoretic logic..- 5.4. Geometry and Group Theory.- Modern geometry deals with the invariants under groups of point transformations. This has produced a very rich geometry of the plane and is highly significant in modern physics..- 5.5. Arithmetic.- Many systems of representing numbers have been used, including hexagesimal, decimal, and binary systems..- 5.6. The Celestial Sphere.- The great distances of the stars make them appear to have fixed positions on a celestial sphere with a diurnal rate of rotation. Various coordinate systems, called equatorial and zodiacal coordinates, have been introduced on this sphere..- 5.7. The Motion of the Sun.- The apparent motion of the sun on the celestial sphere is due to the orbital motion of the earth and is effectively expressed by a Fourier series..- 5.8. Synodic Periods.- Relative planetary positions are approximately periodic..- 5.9. Babylonian Tables.- The first formulation of quantitative astronomy appeared in Babylonia in the form of tables of celestial positions..- 5.10. Geometric Formulations.- The classical description of astronomical motion was geometric and compounded out of uniform circular motions..- 5.11. Astronomical Experience in Terms of Accuracy.- The development of theoretical understanding and of experimental precision were interrelated and complementary..- 5.12. Optical Instruments and Developments.- Modern understanding based on a tremendous range of experimental techniques extends far beyond the solar system..- Exercises.- References.- 6. Natural Philosophy.- 6.1. Analysis.- The extended algebra developed during the Renaissance combined with geometry yielded procedures for finding volumes, areas, and other quantities..- 6.2. The Calculus.- Newton introduced the concepts of the calculus in terms of an axiomatic discussion of motion. Leibnitz used a philosophical notion of infinite and infinitesimal. Both perfected previous algebraic procedures for finding tangents and introduced antidifferentiation instead of algebra for integrals..- 6.3. The Transformation of Mathematics.- The logical inadequacies of the early forms of calculus forced a transformation of analysis into a set-theoretic form in which “intuitive concepts,” i.e., not set-theoretic, were eliminated..- 6.4. The Method of Fluxions.- Infinitesimal analysis dealt with general curves and surfaces. Problems in mechanics were expressed as systems of differential equations..- 6.5. The Behavior of Substance in the Eulerian Formulation.- The behavior of a substance can be described by functions of space and time. Observable information corresponds to volumetric or surface integrals. In particular, the flow of substance through a surface has an integral expression..- 6.6. The Generalized Stokes’ Theorem.- The theorems of Gauss and Stokes are used to transform empirical relations between quantities expressed as integrals into partial differential equations. This procedure is basic for the theory of electricity and magnetism..- 6.7. The Calculus of Variations.- The calculus of variations dealt with extremal properties of functions rather than numbers and opened up new avenues of analysis..- 6.8. Dynamics.- The reformulation of mechanics in terms of variational principles permitted the use of general coordinate systems and was the basis for theoretical developments..- 6.9. Manifolds.- Manifolds are general geometric objects with an infinitesimal structure having a linear geometry..- 6.10. The Weyl Connection.- The affine connection was a further development..- 6.11. The Riemannian Metric.- A Riemannian metric implies an affine connection and a notion of geodesic length. These ideas represent a conceptual frame for theoretical developments..- Exercises.- References.- 7. Energy.- 7.1. The Motion of Bodies.- For a body that approximately retains its shape one can resolve its motion into a gross motion and a local one..- 7.2. The Stress Tensor.- The contact forces within a body are expressible by the symmetric matrix a, which is the original “tensor.”.- 7.3. Deformation and Stress.- The deformation of a substance is given by the matrix B = (JJ’)1/2, where J is the Jacobian matrix of the transformation that describes the deformation. If there is an elastic energy function, B can be related to the tensor a experimentally..- 7.4. An Elastic Collision.- Two steel bars collide. An integral relation does not require second derivatives..- 7.5. Thermodynamic States and Reversibility.- Pressure is a special case of stress. Thermodynamics deals with idealized experiments that can be associated with infinitesimal analysis..- 7.6. Thermodynamic Functions.- The relation between heat and mechanical work is established by means of two thermodynamic functions, the internal energy U and the entropy S..- 7.7. The Carnot Cycle and Entropy.- The second law of thermodynamics represents a fundamental limitation on the conversion of heat into other forms of energy..- 7.8. The Relation with Applied Mathematics.- The expansion of mathematics, represented by infinitesimal analysis, complemented the development of dynamics, electromagnetism, and thermodynamics..- Exercises.- References.- 8. Probability.- 8.1. The Development of Probability.- Probability represented a mathematical description of a new range of phenomena..- 8.2. Applications.- Probability plays an essential role in scientific theories, experimental decision theory, and the theory of games..- 8.3. Probability and Mechanics.- A dynamic motion description and a probability distribution on phase space are complementary. The probability distribution is determined by energy equilibrium requirements..- 8.4. Relation to Thermodynamics.- Statistical mechanics permits the computation of the thermodynamic functions in terms of the fine structure of substance..- 8.5. The Fine Structure of Matter.- The classical concepts of dynamics proved inadequate to cope with the range of newly discovered phenomena and was replaced by quantum mechanics..- 8.6. Analysis.- The needs of applied mathematics spurred the development of analysis through infinite power series, differential equations, and more abstract operator and space theory..- Exercises.- References.- 9. The Parado.- 9.1. Intellectual Ramifications.- With the development of quantum mechanics, the mathematical understanding of natural phenomena in our immediate environment appears to be complete in principle..- 9.2. The Paradox.- But a deterministic mathematical description of phenomena that includes ourselves precludes the control of experiment. No available explanation of this situation appears satisfactory..- 9.3. Final Comment.- Exercises.- References.