Chapter 11

प्रस्तावना

The preface introduces the textbook 'Numerical Analysis and Vector Calculus,' designed according to the syllabus of Vardhman Mahaveer Open University, Kota, for B.Sc. Second Year Mathematics (Paper III). The book is structured to facilitate both teaching and self-study. Recognizing the nature of mathematics, appropriate use of English terminology is made throughout the text. Each unit is authored by subject experts, and standard references have been used to ensure accuracy and depth. The preface acknowledges these contributors and expresses gratitude. The book is also intended to be a valuable resource for students preparing for competitive examinations.

• The book is tailored for B.Sc. Second Year Mathematics, Paper III.
• English mathematical terms are used appropriately.
• Each unit is written by expert mathematicians.
• Standard references have been used for factual accuracy.
• The book is useful for both university exams and competitive exams.
• 📌 Numerical Analysis: The branch of mathematics that deals with finding numerical solutions to mathematical problems.
• 📌 Vector Calculus: The field of mathematics concerned with differentiation and integration of vector fields.

इकाई-1 : परिमित अन्तर (Finite differences)

This section outlines the structure of Unit 1, which focuses on finite differences. The unit covers the following topics: objectives, introduction, forward differences and their operators, forward difference tables, properties of forward difference operators, backward differences and their operators, backward difference tables, properties of backward difference operators, forward difference formula, the fundamental theorem of difference calculus, shift operator (E), properties of the shift operator, differential operator (D), relationships among operators Δ, ∇, E, and D, finding unknown terms in a function with equally spaced values, summary, glossary, answers to self-assessment questions, and practice questions.

• Unit 1 covers finite differences and related operators.
• Topics include forward and backward differences, shift and differential operators.
• The unit includes both theory and practical applications.
• Self-assessment and practice questions are provided.
• 📌 Finite Difference: The difference between function values at specific points, used for numerical analysis.
• 📌 Forward Difference (Δ): An operator that gives the difference between a function value and its next value.
• 📌 Backward Difference (∇): An operator that gives the difference between a function value and its previous value.