Course info

Module – I                                                                                                                        (21 hrs)

          : Vector Functions - Vector Valued Functions, Limits, Continuity, and Derivatives, Geometric Interpretation of r^′(t), Higher Order Derivatives, Integrals of Vector                                Functions, Length of a Space Curve, Arc Length as a Parameter

          : Motion on a Curve - Velocity and Acceleration, Centripetal Acceleration, Curvilinear Motion in the Plane

          : Curvature and components of Acceleration  - definition, Curvature of a Circle, Tangential and Normal Components of Acceleration, The Binormal, Radius of Curvature

          : Partial Derivatives - Functions of Two Variables, Level Curves, Level Surfaces, Higher Order and Mixed Derivatives, Functions of Three or More Variables, Chain Rule, Generalizations

          : Directional Derivative - The Gradient of a Function, A Generalization of Partial Differentia- tion, Method for Computing the Directional Derivative, Functions of Three Variables, Maximum Value of the Directional Derivative, Gradient Points in Direction of Most Rapid Increase of f

          : Tangent planes and Normal Lines - Geometric Interpretation of the Gradient, Tangent Plane, Surfaces Given by z = f (x, y), Normal Line

Module – II                                                                                                                        (24 hrs

             : Curl and Divergence - Vector Fields, definition of curl and divergence, Physical Interpreta- tions

          : Line Integrals - definition of smooth closed and simple closed curves, Line Integrals in the Plane, Method of Evaluation - curve as explicit function and curve given parametrically, Line Integrals in Space, Method of Evaluation, Work, Circulation

          : Independence of Path - Conservative Vector Fields, Path Independence, A Fundamental The- orem, definition of connected, simply connected and multiconnected regions, Integrals Around Closed Paths, Test for a Conservative Field, Conservative Vector Fields in 3-Space, Conservation of Energy

           : Double Integral - Integrability, Area, Volume, Properties, Regions of Type I and II, Iterated Integrals, Evaluation of Double Integrals (Fubini theorem), Reversing the Order of Integration, Laminas with Variable Density-Center of Mass, Moments of Inertia, Radius of Gyration

                        : Double Integrals in  Polar  Coordinates - Polar Rectangles, Change of Variables:  Rectan- gular to Polar Coordinates.

              : Green’s Theorem - Line Integrals Along Simple Closed Curves,  Green’s  theorem  in  plane, Region with Holes

            : Surface Integral - Surface Area, Differential of Surface Area, Surface Integral, Method of Evaluation, Projection of S into Other Planes, Mass of a Surface, Orientable Surfaces, Integrals of Vector Fields-Flux.

          : Stokes’s Theorem - Vector Form of Green’s Theorem, Green’s Theorem in 3-Space Stoke’s Theorem, Physical Interpretation of Curl

Module – III                                                                                                                        (21 hrs)

                    : Triple Integral - definition, Evaluation by Iterated Integrals, Applications, Cylindrical Co- ordinates, Conversion of Cylindrical Coordinates to Rectangular Coordinates,               Conversion of Rectangular Coordinates to Cylindrical Coordinates, Triple Integrals in Cylindrical Coordinates, Spherical Coordinates, Conversion of Spherical                               Coordinates to Rectangular and Cylindrical Co- ordinates, Conversion of Rectangular Coordinates to Spherical Coordinates, Triple Integrals in Spherical Coordinates

            : Divergence Theorem - Another Vector Form of Green’s Theorem, divergence or Gauss’ the- orem, (proof omitted), Physical Interpretation of Divergence

             : Change of Variable in Multiple Integral - Double Integrals, Triple Integrals

   17.1: Complex Numbers - definition, arithmetic operations, conjugate, Geometric Interpretation 17.2: Powers and roots - Polar Form, Multiplication and Division, Integer Powers of z, DeMoivre’s

Formula, Roots

            : Sets in the Complex Plane - neighbourhood, open sets, domain, region etc.

            Functions  of  a  Complex  Variable - complex functions,  Complex Functions as Flows,  Limits and Continuity, Derivative, Analytic Functions-entire functions

  : Cauchy Riemann Equation - A Necessary Condition for Analyticity, Criteria for analyticity, Harmonic Functions, Harmonic Conjugate Functions,

  : Exponential and Logarithmic function - (Complex) Exponential Function, Properties, Pe- riodicity, (‘Circuits’ omitted), Complex Logarithm-principal value, properties, Analyticity

  : Trigonometric and Hyperbolic functions - Trigonometric Functions, Hyperbolic Functions, Properties-Analyticity, periodicity, zeros etc.

Module – IV                                                                                                                        (14 hrs)