1Principles of Mathematical Modeling 1.1How to develop a mathematical model 1.1.1A simple mathematical model 1.1.2Use of a computer 1.1.3Development of mathematical models 1.2Types of models 1.3Stability of models 1.4Dimension, units, and scaling 1.4.1Dimensional analysis 1.4.2Scaling 2Numerical and symbolic computations 2.1Numerical and symbolic computations of derivatives and inte¬grals 2.2Iterative methods 2.3Newton's method 2.4Method of successive approximations 2.5Banach Fixed Point Theorem 2.6Why is it difficult to numerically solve some equations? II Basic Applications 3Application of calculus to classic mechanics 3.1Mechanical meaning of the derivative 3.2Interpolation 3.3Integrals 3.4Potential energy 4Ordinary differential equations and their applications 4.1Principle of transition for ODE 4.2Radioactive decay 4.3Logistic differential equation and its modifications 4.3.1Logistic differential equation 4.3.2Modified logistic equation 4.3.3Stability analysis 4.3.4Bifurcation 4.4Time delay 4.5Approximate solution to differential equations 4.5.1Taylor approximations 4.5.2Pade approximations 4.6Harmonic oscillation 4.6.1Simple harmonic motion 4.6.2Harmonic oscillator with friction and exterior forces 4.6.3Resonance 4.7Lotka-Volterra model 4.8Linearization 5Stochastic models 5.1Method of least squares 5.2Fitting 5.3Method of Monte Carlo 5.4Random walk 6One-dimensional stationary problems 6.1ID geometry 6.2Second order equations 6.3ID Green's function 6.4Green's function as a source 6.5The (5-function III Advanced Applications 7Vector analysis 7.1Euclidean space R3 7.1.1Polar coordinates 7.1.2Cylindrical coordinates 7.1.3Spherical coordinates 7.2Scalar, vector and mixed products 7.3Rotation of bodies 7.4Scalar, vector and mixed product in Mathematica 7.5Tensors 7.6Scalar and vector fields 7.6.1Gradient 7.6.2Divergence 7.6.3Curl 7.6.4Formulae for gradient, divergence and curl 7.7Integral theorems 8Heat equations 8.1Heat conduction equations 8.2Initial and boundary value problems 8.3Green's function for the ID heat equation 8.4Fourier series 8.5Separation of variables 8.6Discrete approximations of PDE 8.6.1Finite-difference method 8.6.2ID finite element method 8.6.3Finite element method in R2 8.7Universality in Mathematical Modeling. Table 9Asymptotic methods in composites 9.1Effective properties of composites 9.1.1General introduction 9.1.2Strategy of investigations 9.2Maxwell's approach 9.2.1Single-inclusion problem 9.2.2Self consistent approximation 9.3Densely packed balls 9.3.1Cubic array 9.3.2Densely packed balls and Voronoi diagrams 9.3.3 Optimal random packing