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Mathematics I

Mathematics I. Instructor: Prof S. K. Ray, Department of Mathematics and Statistics, IIT Kanpur.

1. Calculus of Functions of One Variable
Real numbers, Functions, Sequences, Limit and Continuity, Differentiation : review, successive differentiation, chain rule and Leibnitz theorem, Rolle's and Mean Value Theorems, Maxima/ Minima, Curve sketching, Linear and quadratic approximations, Error estimates, Taylor's theorem, Newton and Picard methods, The Riemann integral, Approximate integration, Natural logarithm, Exponential function, Relative growth rates, L'Hospital's rule geometric applications of integrals, Infinite series, Tests of convergence, Absolute and conditional convergence, Taylor and maclaurin series.

2. Calculus of Functions of Several Variables
Scalar fields, Limit and continuity, Partial derivatives, Chain rules, Implicit differentiation, Directional derivatives, Total differential, Tangent planes and normals, Maxima, Minima and Saddle Points, Constrained maxima and minima, Double Integrals, Applications to Areas and Volumes, Change of variables.

3. Vector Calculus
Vector fields, divergence and curl, Line integrals, Green's theorem, Surface integrals, Divergence theorem, Stoke's theorem and application.
(from nptel.ac.in)

Real Numbers


Lecture 01 - Real Numbers
Lecture 02 - Sequences I
Lecture 03 - Sequences II
Lecture 04 - Sequences III
Lecture 05 - Continuous Functions
Lecture 06 - Properties of Continuous Functions
Lecture 07 - Uniform Continuity
Lecture 08 - Differentiable Functions
Lecture 09 - Mean Value Theorem (One Variable)
Lecture 10 - Maxima/ Minima (One Variable)
Lecture 11 - Taylor's Theorem
Lecture 12 - Curve Sketching
Lecture 13 - Infinite Series I
Lecture 14 - Infinite Series II
Lecture 15 - Test of Convergence
Lecture 16 - Power Series
Lecture 17 - Riemann Integral
Lecture 18 - Riemann Integrable Function
Lecture 19 - Applications of Riemann Integral
Lecture 20 - Length of a Curve
Lecture 21 - Line Integrals
Lecture 22 - Functions of Several Variables
Lecture 23 - Differentiation
Lecture 24 - Derivatives
Lecture 25 - Mean Value Theorem (Multivariables)
Lecture 26 - Maxima/ Minima (Multivariables)
Lecture 27 - Method of Lagrange Multipliers
Lecture 28 - Multiple Integrals
Lecture 29 - Surface Integrals
Lecture 30 - Green's Theorem
Lecture 31 - Stokes' Theorem
Lecture 32 - Gauss' Divergence Theorem

References
Mathematics I
Instructor: Prof S. K. Ray, Department of Mathematics and Statistics, IIT Kanpur. Calculus of Functions of One Variable. Calculus of Functions of Several Variables. Vector Calculus.