| LECTURE | SUMMARY |
|---|---|
| R1 | Read the textbook as we advance |
| R2 | Taylor Series in Two Variables |
| R3 | Discussions of physical meaning of applications of divergence theorem and of Stoke's theorem |
| R4 | Eigen Values and Vectors and their applications |
| R5 | Contravariant and Covariant Vectors | Tensors |
| R6 | Read the chapter on Normal Modes |
| R7 | Metric Tensor | General Coordinate Transformations | Derivative of Basis Vectors & Christoffel Symbols | Geodesics |
| 01 | Preliminaries | Taylor Series | Maclaurin Series | Binomial Theorem |
| 02 | Complex Numbers | Argand Diagram | Euler's Formula | Conjugate | Complex Equations | De Moivre's Theorem | Roots of Complex Numbers |
| 03 | Hyperbolic Functions | Their Use | Logarithms of Complex Numbers | Introduction to Mathematica | Partial Differentiation | Applications of Partial Differentiation | Total Differential & Derivative | Maxima & Minima of Surfaces |
| 04 | Stationary Values of Many Variable Functions | Envelopes |
| 05 | Multiple Integrals & Jacobians |
| 06 | Vectors - preliminaries | Vector Operations | Vector Equations of Surfaces and Lines |
| 07 + 08 + 09 | Vector Spaces | Basis Vectors | Metric Tensor | Inner Product | Linear Operators | Metrices | Complex & Hermitian Conjugate and Trace of a Matrix Differentiation of Vectors | Basis Vectors at a point on a Curve | Velocity and Acceleration in Polar Coordinates | Motion in a Radial Force Field | Basis Vectors & Metric Tensor | Acceleration in Generalized Coordinates | Geodesic Equation |
| 10 + 11 | Curves in Space | Tangent Vectors and Planes | Curvature and Radius of Curvature | Coordinate Surfaces and Curves | Basis Vectors | Del or Nabla Operator | Directional Derivative | General Curvilinear Coordinate Systems | Natural Basis and Dual Basis | Contravariant and covariant Vectors |
| 12 + 13 | Equation of Continuity | Formal Definitions of Divergence and of Curl | Applications of Divergence Theorem | Gauss's Law and Stoke's Theorem and Their Applications | Curl and Rotation of Rigid Bodies | Irroratational and Rotational Fluids |
| 14 | Maxwell Equations and Derivation of Gausass's Law and of Ampere's Law | Further Investigations on Stoke's Law | Metric tensor of the cosmos | Physics of the Robertson - Walker metric |
| 15 | Simple Harmonic Motion | Waves and their Propagation | Average Value of a Function | Odd & Even Functions | Fourier Series | Dirichlet Theorem | Complex Form of Fourier Series |
| 16 + 17 | Fourier Series for General Intervals | Various Applications of Complex Form of Fourier Series | Applications of Fourier Series to Sound | Applications of Fourier Series to Thermodynamics | Relative Intensities of Fourier Harmonics | Parseval's Theorem |
| 19 + 20 | First and Second Order Differential Equations | Applications to Vibrations and Oscillations | Solving Numerical Differential Equations with Mathematica |
| 21 | Legendre Equation | Generating Function | Applications of Legendre Functions | Rodrigues' Formular for Legendre Functions |
| 22 | Laplace Equation | Separation of Variables & Solutions in Spherical coordinators | Applications of Legendre Polynomials in Expansion of Potentials in Electrostatics and Gravitation |
| 23 | Further applications of Laplace Equation | Methods of obtaining solutions for potentials in electrostatics and gravitation for a number of boundary value problems |
| 24 | Gamma Function | Bessel Equation | Bessel Function | Neumann Function | Recursion Relations | Zeros of Bessel Functions | Asymptotic Formulae for Bessel Functions |
| 25 | Wave Equation | Helmholtz Equation | General Solution to Wave Equation in polar coordinates | Calculation of frequency of modes of vibrations of a circular drum skin (Tabla skin) |
| 00000000000 | Topics Covered in 2004 |
| 04 | Matrices | Row Reduction | Rank of a Matrix | Determinants | Gaussian Development | Cramer's Rule |
| 05 | Vectors | Dot & Cross Products | Straight Lines & Planes |
| 06 | Matrix Operations | Rotation Matrix | Matrix Inversion | Inconsistent & Dependent & Homogeneous Equations | Partial Differentiation | Total Differential | Chain Rule |
| 07 | Calculation of Uncertainties | More on Chain Rule | Leibniz Rule to Differentiate Integrals |
| 08 | Applications of Integrals | Double and Triple Integrals | Volumes and Surface Areas | Arc Lengths |
| 09 | Jacobians of Coordinate Transformations | Polar, Spherical, & Cylindrical Coordinates | Applications of Triple Integrals |
| 10 | Applications of Vectors | Triple (Scalar & Vector) Products & Applications | Differentiation of Vectors | Directional Derivatives & Gradients; Gradients in Various Coordinate Systems |
| 11 | Divergence Curl | Laplacian | Line Integrals | Conservative Fields |
| 12 | Green's Theorem | Divergence Theorem in 2D and in 3D | Stoke's Theorem | Physical Interpretation of Divergence |
| 13 | Equation of Continuity | Formal Definitions of Divergence and of Curl | Applications of Divergence Theorem |
| 14 | Gauss's Law and Stoke's Theorem and Their Applications | Curl and Rotation of Rigid Bodies | Irroratational and Rotational Fluids |
| 15 | Maxwell Equations and Derivation of Gausass's Law and of Ampere's Law | Further Investigations on Stoke's Law |
| 16 | Simple Harmonic Motion | Waves and their Propagation | Average Value of a Function | Odd & Even Functions |
| 17 | Fourier Series | Dirichlet Theorem | Complex Form of Fourier Series |
| 18 | Fourier Series for General Intervals | Various Applications of Complex Form of Fourier Series |
| 19 | Applications of Fourier Series to Sound | Applications of Fourier Series to Thermodynamics | Relative Intensities of Fourier Harmonics | Parseval's Theorem |
| 20 | First and Second Order Differential Equations | Applications to Vibrations and Oscillations | Solving Numerical Differential Equations with Mathematica |
| 21 | General Coordinate Transformations | Orthogonal Transformations | Eigen Vectors and Eigen Values | Diagonalization of a Matrix | Applications of Eigen Vectors and Eigen Values |
| 22 | Series Solutions of Differential Equations | Method of Frobenius | Numerical Integration of Differential Equations and their Applications in Physics and Astronomy |
| 23 | Legendre Equation and Functions | Generating Function for Legendre Polynomials | Leibniz's Rule for Differentiation | Rodrigues' Formula | Applications of Legendre Polynomials in Expansion of Potentials in Electrostatics and Gravitation |
| 24 | Gamma Function | Bessel Equation | Bessel Function | Neumann Function | Recursion Relations | Zeros of Bessel Functions | Asymptotic Formulae for Bessel Functions |
| 25 | Applications of Bessel Functions | Theory of Lengthening Pendulum | Derivatives of Bessel Functions |
| 26 | Expansion of Gravitational Potentials | Potential of the Electric Quadrupole | Heights of Columns and Bessel Functions | Bessel Functions in Physical Optics | Resolving Power of Astronomical Telescopes and Bessel Functions |