Chapter 3

प्रस्तावना

The preface introduces the book 'Numerical Analysis and Vector Calculus', designed according to the syllabus of Vardhman Mahaveer Open University, Kota. It is intended for B.Sc. Part II Mathematics students, specifically for the third paper. The book uses appropriate English terminology, considering the nature of mathematics. Various units have been authored by subject experts, and standard references have been used to ensure factual accuracy. The preface also acknowledges the contributions of these authors. The book is also suitable for students preparing for competitive examinations, providing proper guidance in the field of numerical analysis and vector calculus.

• The book follows the syllabus of Vardhman Mahaveer Open University, Kota.
• It is intended for B.Sc. Part II Mathematics, Paper III.
• English mathematical terminology is used throughout.
• Units are written by subject experts using standard references.
• The book is useful for competitive exam preparation.
• 📌 Numerical Analysis: A branch of mathematics focused on obtaining numerical solutions to mathematical problems.
• 📌 Vector Calculus: The study of calculus with vector fields and functions.

इकाई-1 : परिमित अन्तर (Finite differences)

This section outlines the structure of Unit 1, which is dedicated to the concept of finite differences. The unit is divided into several subtopics, including the objectives, introduction, forward and backward difference operators, their properties, difference tables, fundamental theorems, shift operators, and their interrelations. The unit also covers the application of finite differences to estimate unknown values of a function when its values are known at equally spaced points. The content is systematically organized to facilitate step-by-step understanding, starting from basic definitions to advanced applications.

• Unit 1 focuses on finite differences and their applications.
• It covers forward and backward difference operators and their properties.
• The unit explains how to use finite differences to estimate unknown function values.
• Shift and differential operators are also discussed.
• The structure includes objectives, definitions, theorems, and exercises.
• 📌 Finite Difference: The difference between function values at discrete points.
• 📌 Forward Difference Operator: An operator used to compute the difference between successive values of a function.
• 📌 Backward Difference Operator: An operator used to compute the difference between a value and its preceding value.