The course “Multivariate Calculus” is proposed for B.Sc. (Hons) Mathematics as the Core course with 5 credits. The course starts with description of functions of several variables. The course is also beneficial for all students of Physics, Chemistry, Computer Science and Information Technology. After the successful completion of the course the students will be familiar with the following concepts: Limit and continuity of functions of two variables, Partial differentiation, total differentiability and differentiability, sufficient condition for differentiability. Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes, Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems, Definition of vector field, divergence and curl, Double integration over rectangular region, double integration over non-rectangular region, Double integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelepiped and solid regions, Volume by triple integrals, cylindrical and spherical co-ordinates, Change of variables in double integrals and triple integrals. Line integrals, Applications of line integrals: Mass and Work. Fundamental theorem for line integrals, conservative vector fields, independence of path, Green’s theorem, surface integrals, integrals over parametrically defined surfaces. Stoke’s theorem, and the Divergence theorem.
Overview
Syllabus
Week 1Day 1: Module 1 - CylindersDay 2: Module 2 - Quadric SurfacesDay 3: Module 3 - Functions of Several Variables (Domain, Range, and Graph of Functions of Several Variables)Day 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 1
Week 2Day 1: Module 4 - Graph, Level Curves, and Contour of Functions of More Than One VariableDay 2: Module 5 - Level Curves and Contour Lines of Functions of Several Variables AND Limit Along Curves of Functions of Two VariablesDay 3: Module 6 - Limit and Continuity of Functions of Several VariablesDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 2
Week 3Day 1: Module 7 - Partial DifferentiationDay 2: Module 8 - Partial Differentiation of Higher OrderDay 3: Module 9 - Differentiability and Linearization of Functions of Two VariablesDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 3
Week 4Day 1: Module 10 - Total Differentials of Functions of Two Variables AND Differentiability, Linearization, and Total Differential of Functions of Three VariablesDay 2: Module 11 - Chain Rules for Functions of Several VariablesDay 3: Module 12 - Applications of the Chain Rule - Implicit DifferentiationDay 4: DiscussionDay 5: Descriptive Assignment - Assignment 4
Week 5Day 1: Module 13 - Directional Derivative of Functions of Two VariablesDay 2: Module 14 - Directional Derivative of Functions of Three VariablesDay 3: Module 15 - Extreme Values of Functions of Several Variables and Saddle PointsDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 5
Week 6Day 1: Module 16 - Absolute Maxima and Minima on Closed Bounded RegionsDay 2: Module 17 - Lagrange Multipliers with One ConstraintDay 3: Module 18 - Lagrange Multipliers with Two ConstraintsDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 6
Week 7Day 1: Module 19 - Double Integration over Rectangular RegionsDay 2: Module 20 - Double Integration in Non-Rectangular RegionsDay 3: Module 21 - Double Integrals by Reversing the Order of Integration and Area by Double IntegralsDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 7
Week 8Day 1: Module 22 - Double Integration in Polar CoordinatesDay 2: Module 23 - Triple Integration in Cartesian Coordinates, Triple Integrals over General Surfaces in Cartesian Coordinates AND Triple Integrals in Cartesian Coordinates (Changing the Order of Integration)Day 3: Module 24 - Cartesian and Cylindrical Coordinate SystemsDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 8
Week 9Day 1: Module 25 - Spherical Coordinate System in Three Dimensional SpaceDay 2: Module 26 - Triple Integrals in Cylindrical CoordinatesDay 3: Module 27 - Triple Integrals in Spherical CoordinatesDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 9
Week 10Day 1: Module 28 - Change of Variables in Double and Triple Integrals - JacobiansDay 2: Module 29 - Arc Length - Arcs Given by Vector Valued Functions AND Arc Length Parameter, Speed, and Unit Tangent VectorDay 3: Module 30 - Line Integrals - Integration Along CurvesDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 10
Week 11Day 1: Module 31 - Evaluation of Line IntegralsDay 2: Module 32 - Applications of Line Integrals - Finding Centre of MassDay 3: Module 33 - Vector Valued Functions, Vector Fields, Divergence, and CurlDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 11
Week 12Day 1: Module 34 - Line Integral of Vector Valued FunctionsDay 2: Module 35 - The Work Done by a Force Over a Curve in SpaceDay 3: Module 36 - Path Independence of Line Integrals - Fundamental Theorem of Line IntegralsDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 12
Week 13Day 1: Module 37 - The Divergence of a Vector Field AND k-Component of Curl of a Vector FieldDay 2: Module 38 - Green’s Theorem - Normal FormDay 3: Module 39 - Green’s Theorem - Tangential FormDay 4: Interaction based on the three modules coveredDay 5: Objective Assignment - Assignment 13
Week 14Day 1: Module 40 - Surface Area of Surfaces in Cartesian FormDay 2: Module 41 - Surface Integrals when Surfaces are given by Cartesian EquationsDay 3: Module 42 - Orientation of SurfacesDay 4: Interaction based on the three modules coveredDay 5: Descriptive Assignment - Assignment 14
Week 15Day 1: Module 43 - Flux of a Vector Field through a SurfaceDay 2: Module 44 - Parametrized SurfacesDay 3: Module 45 - Surface Area of Parametrized SurfacesDay 4: Module 46 - Surface Integral of Parametrized SurfacesDay 5: Interaction based on the three modules coveredDay 6: Objective Assignment - Assignment 15
Week 16Day 1: Module 47 - Stokes’ Theorem Part ADay 2: Module 48 - Stokes’ Theorem Part BDay 3: Module 49 - The Divergence Theorem of GaussDay 4: Module 50 - The Divergence Theorem for General RegionsDay 5: Interaction based on the three modules coveredDay 6: Descriptive Assignment - Assignment 16
Taught by
Dr. Bijumon Ramalayathil
