Part 1: Vector Fields with applications to Thermodynamics and Irreversibility. 1. Classes of Equations and Similarity Solutions. 1.1. Hierarchy of Partial Differential Equations. 1.2 General Integral and Arbitrary Functions. 1.3. Unforced P.D.E. with First-Order Derivatives. 1.4. Quasi-Linear and Forced First-Order P.D.E.S. 1.5. Differentials of First-Degree in Three Variables. 1.6. P.D.E.s with Constant Coefficients and All Derivatives of Same Order. 1.7. Harmonic and Biharmonic Functions on the Plane. 1.8. Forced Linear P.D.E. with Derivatives of Constant Order. 1.9. Forced Harmonic and Biharmonic Equations. 1.10. Conclusion. 2. Thermodynamics, Irreversibility, Compressible Flow and Shocks. 2.1. Work, Heat, Entropy and Temperature. 2.2. Functions of State and Constitutive Properties. 2.3. Three Principles and Four Processes of Thermodynamics. 2.4. Entropy Production and Diffusive Properties. Part 2: Compressible Flow with applications to Engines, Shocks and Nozzles. 2. Thermodynamics, Irreversibility, Compressible Flow and Shocks. 2.5 Equation of State and Thermodynamic Cycles. 2.6 Adiabatic Compressible Fluid Flow. 2.7. Vortex Sheet and the Normal Shock (Rankine 1870; Huginot 1887). 2.8. Oblique Shock (Busemann 1931) and Adiabatic Turn (Prandtl, Meyer 1908). 2.9. The ‘Choked’ or ‘Shocked’ Nozzle. 2.10. Conclusion.