Vector Calculus: scalar and vector fields – level surfaces – directional derivatives, gradient, curl, divergence – Laplacian – line and surface integrals – theorems of Green, Gauss, and Stokes.

Sequences and Series of Functions: complex sequences – sequences of functions – uniform convergence of series – test for convergence – uniform convergence for series of functions.

Mathematical review and computer arithmetic – numbers and errors; Nonlinear equations; Direct methods for linear systems; Iterative Methods for Linear Systems; Eigenvalues and Eigenvectors – power method, inverse power method, QR method; Approximation Theory – norms, orthogonaliza- tion, polynomial approximation, piecewise polynomial approximation, trigonometric approximation, rational approximation, wavelet bases; Numerical Differentiation; Numerical Integration- Romberg Integration, Gauss Quadrature, Adaptive Quadrature; Numerical Ordinary Differential Equations – single step and multi-st

Vector Space, norm, and angle – linear independence and orthonormal sets – row reduction and echelon forms, matrix operations, including inverses – effect of round-off error, operation counts – block/banded matrices arising from discretization of differential equations – linear dependence and independence – subspaces and bases and dimensions – orthogonal bases and orthogonal projections – Gram-Schmidt process – linear models and least-squares problems – eigenvalues and eigenvectors – diagonalization of a matrix – symmetric matrices – positive definite matrices – similar matrices – linear tr

Introduction: Simulation classification – Objectives, concepts, and types of models – Modeling: 6- DOF models for aerospace vehicle with prescribed control surface inputs – Control Systems: Mechan- ical (structural), Hydraulic systems and their modeling – Block diagram representation of systems – Dynamics of aerospace vehicles – Pilot station inputs – Cues for the Pilot: Visual, biological, and stick force – Virtual Simulation: Fly-by-wire system simulation – Uncertainty Modeling & Simula- tion: Characterization of uncertainty in model parameters and inputs, use of simulation to propaga

Fundamentals of matrix algebra and vector spaces – Solution of simultaneous equations for square, under-determined, and over-determined systems – Concepts of basis vector transformations – Sim- ilarity and adjoint transformations – Eigenvalues and eigenvectors – Jordan form – Characteristic equation – Analytic functions of square matrices and Cayley-Hamilton theorem – Concepts of state, state-space, state-vector – Methods for obtaining the system mathematical model in the state-space form – State-space Form for Aerospace Systems: Aircraft dynamics, missile dynamics, inertial navi- gation sy

Panel methods – Unsteady potential flows – Compressible flow over wings – Axisymmetric flows and slender body theories – Boundary layer analysis – Viscous-inviscid coupling – Flight vehicle aerody- namics – Rotor aerodynamics – Low Reynolds number aerodynamics – Flapping wings – Two- and three-dimensional flow separation.

Introduction to stability – Review of dynamical systems concepts – Instabilities of fluids at rest – Stability of open shear flows: Inviscid and viscous theory, spatio-temporal stability analysis (absolute and convective instabilities) – Parabolized stability equation – Transient growth – Introduction to global instabilities.

Governing equations for viscous fluid flow – Heat conduction and compressibility – Exact solutions – Laminar boundary layer approximations – Similar and nonsimilar boundary layers – Momentum integral methods – Separation of boundary layer – Compressible boundary layer equations – Recov- ery factor – Reynolds analogy – Similar solutions – Stability of boundary layer flows: Transition prediction and bypass transition – Turbulent Flows: Phenomenological theories – Reynolds stress – Turbulent boundary layer – Momentum integral methods – Turbulent free shear layer – Flow separation.

Multidisciplinary Design Optimization (MDO): Need and importance – Coupled systems – Analyser vs. evaluator – Single vs. bi-level optimisation – Nested vs.

General features and applications of high temperature flows – Equilibrium Kinetic Theory: Maxwellian distribution, collision rates and mean free path – Chemical thermodynamics – Mixture of perfect gases, law of mass action – Statistical Mechanics: Enumeration of micro-states, energy distribution, contribution of internal structure – Equilibrium Flow: Ideal dissociating gas, equilibrium shock wave relations, nozzle flows – Vibrational and chemical rate processes – Flows with vibrational and chemical non-equilibrium.