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.

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1. Introduction.- 1.1. Vocational Aspects.- 1.2. Intellectual Attitudes.- 1.3. Opportunities in Applied Mathematics.- 1.4. Course Objectives.- Exercises.- 2. Simulations.- 2.1. Organized Efforts.- 2.2. Staging.- 2.3. Simulations.- 2.4. Influence Block Diagram and Math Model.- 2.5. Temporal Patterns.- 2.6. Operational Flight Trainer.- 2.7. Block Diagrams.- 2.8. Equipment.- 2.9. The Time Pattern of the Simulation.- 2.10. Programming.- 2.11. Management Considerations.- 2.12. Validity.- Exercises.- References.- 3. Understanding and Mathematics.- 3.1. Experience and Understanding.- 3.2. Unit Experience.- 3.3. The Exact Sciences.- 3.4. Scientific Understanding.- 3.5. Logic and Arithmetic.- 3.6. Algebra.- 3.7. Axiomatic Developments.- 3.8. Analysis.- 3.9. Modern Formal Logic.- 3.10. Pure and Applied Mathematics.- 3.11. Vocational Aspects.- Exercises.- References.- 4. Ancient Mathematics.- 4.1. Ancient Arithmetic.- 4.2. Egyptian Mathematics.- 4.3. Babylonian Mathematics.- 4.4. Greece.- 4.5. Euclid's Elements.- 4.6. Magnitudes.- 4.7. Geometry and Philosophy.- 4.8. The Conic Sections.- 4.9. Parabolic Areas.- Exercises.- References.- 5. Transition and Developments.- 5.1. Algebra.- 5.2. Non-Euclidean Geometry.- 5.3. Geometric Developments.- 5.4. Geometry and Group Theory.- 5.5. Arithmetic.- 5.6. The Celestial Sphere.- 5.7. The Motion of the Sun.- 5.8. Synodic Periods.- 5.9. Babylonian Tables.- 5.10. Geometric Formulations.- 5.11. Astronomical Experience in Terms of Accuracy.- 5.12. Optical Instruments and Developments.- Exercises.- References.- 6. Natural Philosophy.- 6.1. Analysis.- 6.2. The Calculus.- 6.3. The Transformation of Mathematics.- 6.4. The Method of Fluxions.- 6.5. The Behavior of Substance in the Eulerian Formulation.- 6.6. The Generalized Stokes' Theorem.- 6.7.The Calculus of Variations.- 6.8. Dynamics.- 6.9. Manifolds.- 6.10. The Weyl Connection.- 6.11. The Riemannian Metric.- Exercises.- References.- 7. Energy.- 7.1. The Motion of Bodies.- 7.2. The Stress Tensor.- 7.3. Deformation and Stress.- 7.4. An Elastic Collision.- 7.5. Thermodynamic States and Reversibility.- 7.6. Thermodynamic Functions.- 7.7. The Carnot Cycle and Entropy.- 7.8. The Relation with Applied Mathematics.- Exercises.- References.- 8. Probability.- 8.1. The Development of Probability.- 8.2. Applications.- 8.3. Probability and Mechanics.- 8.4. Relation to Thermodynamics.- 8.5. The Fine Structure of Matter.- 8.6. Analysis.- Exercises.- References.- 9. The Parado.- 9.1. Intellectual Ramifications.- 9.2. The Paradox.- 9.3. Final Comment.- Exercises.- References.