Course

Core Science - Mathematics I

Indian Institute of Technology Kanpur

Core Science - Mathematics I is a comprehensive course designed for students to master fundamental mathematical concepts. The course is structured as follows:

  1. Calculus of Functions of One Variable: Topics include real numbers, functions, sequences, limits, continuity, differentiation, the Riemann integral, and Taylor's theorem.
  2. Calculus of Functions of Several Variables: Focuses on scalar fields, partial derivatives, maxima/minima, and applications including double integrals.
  3. Vector Calculus: Covers vector fields, line integrals, Green's theorem, surface integrals, and Stokes' theorem.

This course equips students with the skills needed to analyze and solve complex mathematical problems, emphasizing both theoretical and practical applications.

Course Lectures
  • Lecture 1 - Real Number
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces real numbers and their properties, setting the foundation for understanding functions and sequences. Students will explore:

    • Types of real numbers
    • Operations with real numbers
    • The number line and intervals
    • Applications of real numbers in calculus

    Understanding real numbers is crucial for further studies in mathematics, as it leads to a solid grasp of functions and limits.

  • Lecture 2 - Sequences I
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module focuses on the study of sequences, including definitions, types, and convergence criteria. Students will learn:

    • The concept of a sequence and its notation
    • Convergent and divergent sequences
    • Monotonic and bounded sequences
    • Applications of sequences in calculus

    Mastering sequences is essential for understanding limits and functions in calculus.

  • Lecture 3 - Sequences II
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module continues the exploration of sequences, focusing on advanced concepts and tests for convergence. Topics include:

    • Divergence tests
    • Comparison tests
    • Ratio and root tests
    • Power series and their convergence

    Understanding these concepts is vital for analyzing functions and their behavior in calculus.

  • Lecture 4 - Sequences III
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module delves deeper into sequences, examining specific types such as geometric and arithmetic sequences. Key topics include:

    • Definitions and formulas for geometric sequences
    • Characteristics of arithmetic sequences
    • Sum formulas and applications
    • Real-world applications of sequences in various fields

    Understanding these sequences prepares students for more complex functions in calculus.

  • Lecture 5 - Continuous Function
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces continuous functions and their importance in calculus. Key concepts include:

    • Definition of continuous functions
    • Properties of continuity
    • Types of discontinuities
    • Importance of continuous functions in calculus applications

    Understanding continuous functions is critical for exploring limits and derivatives in calculus.

  • Lecture 6 - Properties of Continuous function
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module focuses on the properties of continuous functions, exploring various types and their implications. Students will learn:

    • Intermediate Value Theorem
    • Extreme Value Theorem
    • Uniform continuity
    • Applications of continuous functions in real-world scenarios

    These properties are essential for understanding the behavior of functions in calculus.

  • Lecture 7 - Uniform Continuity
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces the concept of uniform continuity, a stronger form of continuity. Key topics include:

    • Definition and differences from regular continuity
    • Examples of uniformly continuous functions
    • The significance of uniform continuity in calculus
    • Applications in mathematical analysis

    Understanding uniform continuity aids in analyzing function behavior in various contexts.

  • Lecture 8 - Differentiable function
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers differentiable functions, a crucial concept in calculus. Students will learn about:

    • The definition of differentiability
    • Relationship between continuity and differentiability
    • Examples of differentiable functions
    • How to find derivatives using various methods

    Mastering differentiable functions is essential for understanding derivatives and their applications.

  • Lecture 9 - Mean Value Theorems
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module delves into the Mean Value Theorems, which relate derivatives to function behavior. Key topics include:

    • The statement and implications of the Mean Value Theorem
    • Applications in finding function values
    • Geometric interpretation of the theorem
    • Examples and problem-solving techniques

    Understanding these theorems is vital for analyzing function behavior in calculus.

  • Lecture 10 - Maxima - Minima
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module explores the concepts of maxima and minima, critical points, and their significance. Students will learn:

    • How to find critical points
    • The role of the first and second derivative tests
    • Applications of maxima and minima in optimization problems
    • Real-world examples

    Understanding these concepts is essential for solving practical problems using calculus.

  • Lecture 11 -Taylor's Theorem
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces Taylor's Theorem, which provides a way to approximate functions using polynomials. Key topics include:

    • The statement of Taylor's Theorem
    • Applications in approximation of functions
    • Understanding Taylor and Maclaurin series
    • Examples and problem-solving techniques

    Mastering Taylor's Theorem is crucial for analyzing function behavior and approximations in calculus.

  • Lecture 12 - Curve Sketching
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers curve sketching, a vital skill in understanding functions. Students will learn:

    • How to analyze functions for curve sketching
    • Identifying intercepts, asymptotes, and critical points
    • Using first and second derivative tests for sketching
    • Applications in real-world problems

    Curve sketching is essential for visualizing and interpreting function behaviors in calculus.

  • Lecture 13 - Infinite Series I
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces the concept of infinite series, focusing on their definitions and types. Key topics include:

    • Convergent and divergent series
    • Geometric series and their properties
    • Applications of infinite series in calculus
    • Examples and problem-solving techniques

    Understanding infinite series is crucial for further studies in calculus and analysis.

  • Lecture 14 - Infinite Series II
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module continues the exploration of infinite series, examining advanced concepts and convergence tests. Topics include:

    • Comparison tests for convergence
    • Ratio and root tests
    • Convergence of power series
    • Applications in calculus

    Mastering these concepts is essential for analyzing functions and series in higher-level calculus.

  • Lecture 15 - Tests of Convergence
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module delves into tests of convergence, essential for analyzing infinite series. Key topics include:

    • The significance of convergence tests
    • Examples of various tests
    • Application of tests in series analysis
    • Problem-solving techniques

    Understanding convergence tests is vital for working with infinite series in calculus.

  • Lecture 16 - Power Series
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces power series, a special type of infinite series. Students will learn about:

    • The definition and properties of power series
    • Radius and interval of convergence
    • Applications of power series in calculus
    • Examples and problem-solving techniques

    Mastering power series is essential for approximating functions and analyzing their behavior.

  • Lecture 17 - Riemann integral
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers the Riemann integral, a fundamental concept in calculus. Key topics include:

    • The definition of the Riemann integral
    • Properties and applications of the Riemann integral
    • Understanding Riemann sums and their significance
    • Examples and problem-solving techniques

    Understanding the Riemann integral is crucial for further studies in calculus and analysis.

  • Lecture 18 - Riemann Integrable functions
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module explores Riemann integrable functions, focusing on their characteristics and applications. Students will learn:

    • The definition of Riemann integrable functions
    • Conditions for integrability
    • Examples of Riemann integrable and non-integrable functions
    • Applications in calculus and real analysis

    Mastering Riemann integrable functions is essential for understanding integration in calculus.

  • Lecture 19 - Applications of Riemann Integral
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module examines applications of the Riemann integral in various fields. Key topics include:

    • Finding areas under curves
    • Calculating volumes of solids of revolution
    • Applications in physics and engineering
    • Examples and problem-solving techniques

    Understanding these applications is vital for applying calculus concepts in real-world scenarios.

  • Lecture 20 - Length of a curve
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers the length of a curve, an important aspect of calculus. Students will learn about:

    • The definition of curve length
    • Formulas for calculating length
    • Applications in geometry and physics
    • Examples and problem-solving techniques

    Understanding curve length is essential for applications involving curves and their properties in calculus.

  • Lecture 21 - Line integrals
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces line integrals, a fundamental concept in vector calculus. Key topics include:

    • The definition of line integrals
    • Applications in physics and engineering
    • Calculating line integrals along curves
    • Examples and problem-solving techniques

    Mastering line integrals is crucial for understanding vector fields and their applications in calculus.

  • Lecture 22 - Functions of several variables
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module focuses on functions of several variables, expanding the understanding of calculus. Key topics include:

    • The definition of functions of several variables
    • Graphical representation and interpretation
    • Partial derivatives and their significance
    • Applications in optimization problems

    Mastering functions of several variables is essential for advanced studies in multivariable calculus.

  • Lecture 23 - Differentiation
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module explores differentiation in the context of functions of several variables. Students will learn:

    • The concept of partial derivatives
    • Chain rule for functions of several variables
    • Applications in real-world problems
    • Examples and problem-solving techniques

    Understanding differentiation in this context is crucial for advanced calculus applications.

  • Lecture 24 - Derivatives
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers derivatives in the context of functions of several variables. Key topics include:

    • Gradient vectors and their significance
    • Directional derivatives
    • Applications in optimization and real-world scenarios
    • Examples and problem-solving techniques

    Understanding derivatives in this context is essential for advanced studies in multivariable calculus.

  • Lecture 25 - Mean Value Theorem
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module focuses on the Mean Value Theorem for functions of several variables. Key topics include:

    • The statement and implications of the theorem
    • Applications in real-world problems
    • Geometric interpretation of the theorem
    • Examples and problem-solving techniques

    Understanding this theorem is vital for analyzing functions of several variables in calculus.

  • Lecture 26 - Maxima Minima
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module explores maxima and minima for functions of several variables. Key topics include:

    • Finding critical points in multivariable functions
    • Second derivative test for local extrema
    • Applications in optimization problems
    • Examples and problem-solving techniques

    Mastering these concepts is essential for solving practical problems in multivariable calculus.

  • Lecture 27 - Method of Lagrange Multipliers
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces the method of Lagrange multipliers, a technique for finding extrema of functions subject to constraints. Key topics include:

    • The formulation of the Lagrange multiplier method
    • Applications in optimization problems with constraints
    • Examples and problem-solving techniques
    • Geometric interpretation of the method

    Understanding this method is crucial for advanced optimization problems in calculus.

  • Lecture 28 - Multiple Integrals
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers multiple integrals, an extension of the Riemann integral to functions of several variables. Key topics include:

    • Definition and properties of double and triple integrals
    • Applications in calculating areas and volumes
    • Techniques for evaluating multiple integrals
    • Examples and problem-solving techniques

    Mastering multiple integrals is essential for advanced studies in calculus and real analysis.

  • Lecture 29 - Surface Integrals
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module explores surface integrals, focusing on their definitions and applications. Key topics include:

    • The definition of surface integrals
    • Applications in physics and engineering
    • Calculating surface integrals over various surfaces
    • Examples and problem-solving techniques

    Mastering surface integrals is crucial for understanding vector calculus and its applications.

  • Lecture 30 - Green's Theorem
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers Green's Theorem, a fundamental result in vector calculus relating double integrals and line integrals. Key topics include:

    • The statement and implications of Green's Theorem
    • Applications in physics and engineering
    • Examples and problem-solving techniques
    • Geometric interpretation of the theorem

    Understanding Green's Theorem is essential for applying vector calculus in real-world scenarios.

  • Lecture 31 - Stokes Theorem
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module introduces Stokes' Theorem, a powerful result relating surface integrals and line integrals. Key topics include:

    • The statement and implications of Stokes' Theorem
    • Applications in physics and engineering
    • Examples and problem-solving techniques
    • Geometric interpretation of the theorem

    Understanding Stokes' Theorem is crucial for applying vector calculus in various fields.

  • Lecture 32 - Gauss Divergence Theorem
    Prof. Swagato K. Ray, Prof. Shobha Madan, Dr. P. Shunmugaraj

    This module covers the Gauss Divergence Theorem, relating volume integrals and surface integrals. Key topics include:

    • The statement and implications of the Divergence Theorem
    • Applications in physics and engineering
    • Examples and problem-solving techniques
    • Geometric interpretation of the theorem

    Mastering the Divergence Theorem is essential for applying vector calculus principles in real-world applications.