3rd Semester AMPS

Torsion: Torsion of circular cross-section shafts. Non-circular cross-sections. Thin-walled closed-section. Warping function. The membrane analogy. Elastoplastic torsion. Bending: Euler–Bernoulli bending. Asymmetric bending. Eccentric axial loading. Composite sections. Elastoplastic bending. Curved beams. Limitations of Euler-Bernoulli theory. The effect of concentrated load. The cantilever problem. Bending with shear: Shear flow. Composite sections. Thin-walled sections. Shear center. Elastic curve: Methods for calculating deflection by integration. Moment-area method. Solving statically indeterminate problems using the elastic curve. Deflection from shear. Buckling: Euler s critical load. The buckling bifurcation problem. Elastoplastic buckling. Design of columns for central and eccentric load. Energy methods and theorems. Strain energy. Castigliano s Theorems.

Introductory concepts: Temperature, Heat. Laws of dilute gases. Isothermal and Adiabatic transformation. Kinetic theory of gases. Equation of state for gases. First law of Thermodynamics: Work, Internal energy. Reversible and irreversible transformations. Heat capacity, Mayers equation. Enthalpy, heat transfer. Heat engines, Carnot engine. Thermodynamic coefficient μ, Carnot. Second law of thermodynamics: Entropy. Irreversible processes. Carnot theorem, Clausius inequalities. Entropy changes, Tds equations. Thermodynamic Potentials: Legendre transforms. Maxwell equations. Helmholtz, Gibbs functions. Phase changes. Open Systems, Chemical Potential: Equilibrium of phase changes. Chemical reactions. Mixing processes. Third law of thermodynamics: Formulations of the third law of thermodynamics. Cooling methods. Applications of the third law of thermodynamics.

Introduction to Fortran, Matlab and Mathematica. An Introduction to Rounding Errors: Numbers Arithmetic Representation, Basic Rounding Errors. Linear Systems: Gauss Elimination Method, LUandCholeski Factorization Methods. Norms (Vector, Matrix, Function).Stability of Linear Systems.Fixed Point Iteration Schemes.Jacobi, Gauss‐Seidel and Relaxation Methods.Least Squares Methods and Applications.Numerical Computation of Eigenvalues/Eigenvectors.Nonlinear Equations: Bisection Method. Fixed Point Iteration Methods, Newton-Raphson (for Equations and Systems).Interpolation and Approximation: Lagrange Interpolation.Newton Polynomial.Hermite Interpolation.Cubic Splines.Best Approximation via Least Squares. Orthogonal Polynomials.Numerical Integration: Newton‐CotesRules (Trapezoidal, Simpson, 3/8). Gauss Integration.Introduction to ODEs: Euler, Taylor and Runge-Kutta Methods.

Not provided.

Introduction to microeconomic theory. Supply and demand. Equilibrium - Price formation. Theories of demand: Consumer behavior, the theory of absolute utility, the theory of ordinal utility. Theory of production and production costs. Market forms: Perfect competition, monopoly, monopolistic competition, oligopoly.

Simple Harmonic Oscillator (free, damped, forced), [Dumping, Forced oscillations, Complex Impedance Resonance effects]. Coupled Oscillators, [Normal modes, Systems with many degrees of freedom, Periodic system, Electrical Oscillations]. Waves in 1-dimensional continuous media, [Wave-differential equation in 1-dimension, Traveling waves, Standing waves, Reflection, transmission, in discontinuities.] Dispersion, [Wave-packets, Phase- and Group- velocities, Surface waves in liquids.] Fourier methods, [Bandwidth theorem, Finite elastic medium-Fourier series, Infinite elastic medium-Fourier Integral, Bandwidth theorems and Uncertainty Principle]. Waves in 2-dimensional continuous media, [Wave-differential equation in 2-dimensions, Standing waves in 2-D, Degeneracy, Density of States, Traveling waves in 2-D, Reflection-transmission, Wave-guiding.] Waves in 3-dimensional elastic media [(acoustic, electromagnetic). Polarization, Fresnel coefficients, Acoustic waves, Electromagnetic waves, Polarization, Fresnel coefficients.] Interference, Diffraction (general principles), [Wavefront-splitting interference, Amplitude-splitting interference, Fraunhofer diffraction].

Integral Calculus of Multivariable Functions: Double integral: Double integral, Fubini s theorem, change of variables, triple integral, change of variables, applications of double and triple integrals. Line integrals: Line integrals of the first kind and applications, line integrals of the second kind and applications, Green s theorem. Surface integrals: Elements from surface theory, surface area, surface integral of the first kind and applications, surface integral of the second kind and applications. Basic theorems of Vector Analysis: Stokes theorem and applications, Gauss s theorem and applications, special vector fields, integral form of divergence and curl, applications of vector analysis.

Mercantilism and its decline, The heyday of mercantilist theory: Thomas Mun, The reaction against mercantilism: Dudley North, William Petty, David Hume. The Physiocrats, Quesnay s Economic Table, The theoretical legacy of the Physiocrats. Adam Smith: The theory of value, The theory of distribution, The theory of capital and productive labor. David Ricardo: Theory of value, Land rent, Wages and profit. The disintegration of the classical school: Malthus, J.B. Say, The disputes surrounding the theory of value, Senior, Carey & Bastiat, Sismondi, The Utopian Socialists, John Stuart Mill. Karl Marx: Abstract labor and value, The theory of capital, Reproduction of the total system of production, Values and production prices. The neoclassical school: Marginal utility and equilibrium of supply and demand, The production function. The Neo-Ricardian school: Production systems and relative prices, Basic and non-basic commodities, The issue of ""negative prices."" Keynes: The ""General Theory"" and its era, effective demand and the multiplier, long-term expectations and marginal efficiency of capital.

Introductory Concepts: Definitions, Concept of solution and geometric characteristics. Initial-boundary value problems. Well-posed problems. Differential equations with separable variables, linear, homogeneous, exact. First-order differential equations: Riccati, Lagrange, Clairaut. Qualitative Theory: Existence and uniqueness of solution. Picard, Peano Theorems. Linear differential equations: General theory. Linear independence. Wronskian determinant. Homogeneous equations with constant coefficients. Variation of parameters method (Lagrange). Method of undetermined coefficients. Euler equation. Solution by series: Power series. Solution at an ordinary point. Legendre equation. Solution at a regular singular point. Frobenius Theory. Bessel equations. Systems of differential equations: Introduction, solution by elimination. General theory. Systems with constant coefficients, homogeneous and non-homogeneous. Laplace Transform: Definition. Properties. Inverse Laplace transform. Applications. Heaviside function. Dirac delta function. Convolution. Use of computational programs.

The Historic and Systematic Approach to Philosophy. Problems and Periods of Western Philosophy. Systematic presentation and analysis of central issues in Philosophy: the validity of knowledge, truth, mind and matter, language and the real world, the significance of Philosophy today.