Vector and Tensor Analysis

📘 Vector and Tensor Analysis (2026–2027 Edition)

Vector and Tensor Analysis: Vector Analysis, Tensor Calculus, & Mathematical Physics Applications (2026–2027 Edition) is a comprehensive, concept-oriented textbook designed for BS Mathematics students, educators, researchers, and professionals in mathematics, applied mathematics, physics, engineering, and related scientific disciplines. This book provides an in-depth understanding of vector algebra, vector geometry, vector calculus, tensor analysis, curvilinear coordinate systems, integral theorems, and advanced mathematical structures used in modern physical sciences and engineering applications.

This resource is ideal for conceptual understanding, university coursework, competitive exams, mathematical problem solving, research studies, & advanced scientific learning. The book bridges classical vector analysis with modern tensor calculus and geometrical applications, enabling readers to understand multidimensional mathematical systems, coordinate transformations, differential operators, tensor operations, & their applications in physics and engineering. The content emphasizes interdisciplinary integration of pure mathematics, applied mathematics, geometry, calculus, tensor theory, & mathematical physics for higher-level analytical studies.

🧮 Chapter 1: Algebra of Vectors
• Introduction & basics of vectors
• Coordinate systems & unit vectors
• Definitions & vector operations in analytic form
• Dot product & applications
• Cross product & applications
• Scalar triple product
• Vector triple product & vector identities
• Linear dependence & related concepts
• Exercise

📐 Chapter 2: Geometry of Vectors
• Introduction & basics
• Vector equations of lines
• Vector equations of planes
• Vector equation of sphere
• Exercise

📊 Chapter 3: Vector Differentiation and Integration
• Introduction & vector functions
• Vector derivatives
• Applications of derivatives
• Multivariable vector functions
• Vector integration
• Exercise

🌐 Chapter 4: Gradient, Divergence, and Curl
• Introduction to vector fields
• Gradient & derivatives
• Divergence & Laplacian
• Curl & properties
• Vector identities
• Exercise

📘 Chapter 5: Line, Surface, and Volume Integrals and Related Integral Theorems
• Introduction
• Line integrals
• Surface integrals
• Volume integrals & regions
• Fundamental integral theorems
• Advanced integral relations
• Exercise

🧭 Chapter 6: Curvilinear Coordinates
• Basics of curvilinear coordinates
• Rectangular Cartesian coordinates
• Cylindrical coordinate system
• Spherical coordinate system
• Transformation between cylindrical & spherical systems
• Exercise

🧩 Chapter 7: Cartesian Tensors
• Fundamentals of Cartesian tensors
• Basic tensor symbols & operations
• Tensor theory & properties
• Tensor calculus & applications
• Eigenvalues & invariants of tensors
• Exercise

🔬 Chapter 8: General Tensors
• Foundations of tensor analysis
• Fundamental tensor tools
• Classification of tensors
• Transformation laws
• Tensor algebra & operations
• Symmetry in tensors
• Metric tensor & associated structures
• Christoffel symbols & differential relations
• Covariant differentiation
• Geometrical & physical interpretations
• Integral theorems in tensor form
• Riemannian geometry & curvature tensors
• Ricci & Einstein structures
• Advanced tensor relations
• Geodesics & applications
• Exercise

This book is inspired by the authors:
Louis Brand, A. P. French, Pavel Grinfeld, J. L. Synge, A. Schild, D. E. Bourne, Robert C. Wrede, Murray R. Spiegel, Richard L. Bishop, & Harley Flanders.

📲 Download Vector and Tensor Analysis (2026–2027 Edition) to explore vector algebra, tensor calculus, curvilinear coordinates, integral theorems, differential geometry, and advanced mathematical physics concepts. Ideal for BS Mathematics students, educators, researchers, and professionals seeking mastery in vector and tensor analysis.