The mathematical theory of probability has its roots in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the “problem of points”). Christiaan Huygens published a book on the subject in 1657. Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. Fairly quickly this became the mostly undisputed axiomatic basis for modern probability theory but alternatives exist, in particular the adoption of finite rather than countable additivity by Bruno de Finetti. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. In this text, all theorems and axioms have been explained by a large number of solved examples. This text will prove to be useful for undergraduate and postgraduate students of mathematics.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process tointegration. Differentiation has applications to nearly all quantitative disciplines. The subject matter has been discussed in such a simple way that the students will find no difficulty to understand it. The proofs of various principles and examples has been given with minute details.

Dr. Rashmi Rana is associated with Department of Mathematics at Chaudhary Charan Singh University, Meerut (UP). She has written a book on Vedic Mathematics.

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3. The term “vector calculus” is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. The basic objects in vector calculus are scalar fields and vector fields. These are then combined or transformed under various operations, and integrated. This book has been designed as an introductory text which undertakes a through exploration the concepts and practices which define the subject. The book will prove very useful for students and teachers.

The mathematical theory of probability has its roots in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the “problem of points”). Christiaan Huygens published a book on the subject in 1657. Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. Fairly quickly this became the mostly undisputed axiomatic basis for modern probability theory but alternatives exist, in particular the adoption of finite rather than countable additivity by Bruno de Finetti. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. In this text, all theorems and axioms have been explained by a large number of solved examples. This text will prove to be useful for undergraduate and postgraduate students of mathematics.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process tointegration. Differentiation has applications to nearly all quantitative disciplines. The subject matter has been discussed in such a simple way that the students will find no difficulty to understand it. The proofs of various principles and examples has been given with minute details.

Dr. Rashmi Rana is associated with Department of Mathematics at Chaudhary Charan Singh University, Meerut (UP). She has written a book on Vedic Mathematics.

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3. The term “vector calculus” is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. The basic objects in vector calculus are scalar fields and vector fields. These are then combined or transformed under various operations, and integrated. This book has been designed as an introductory text which undertakes a through exploration the concepts and practices which define the subject. The book will prove very useful for students and teachers.