“Vector Analysis, Tensor Analysis and Linear Vector Space” authored by S. P. Kuila (NCBA)

Name: "Vector Analysis, Tensor Analysis and Linear Vector Space" authored by S. P. Kuila (NCBA) - Booksbp.com
SKU: 9789352550333
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SKU: 9789352550333

1. Vector Analysis (Vector Calculus & Algebra)

This section builds on standard vector algebra to develop field theories used across electromagnetism and classical mechanics.

- Vector Algebra: Scalar and vector products (dot and cross products), triple products, and scalar/vector fields.
- Vector Differentiation: Differential operators including Gradient (\(\nabla \phi\)), Divergence (\(\nabla \cdot \mathbf{A}\)), and Curl (\(\nabla \times \mathbf{A}\)).
- Ordinary & Curvilinear Coordinates: Orthogonal curvilinear coordinates, expression of differential operators in cylindrical and spherical coordinate systems.
- Vector Integration: Line integrals, surface integrals, and volume integrals.
- Integral Theorems: Gauss’s Divergence Theorem, Stokes’ Theorem, and Green’s Theorem in a plane.

2. Tensor Analysis

Tensors extend vectors to higher dimensions, enabling the description of physical laws independent of any specific coordinate system.

- Foundations: Coordinate transformations, dummy and free indices, and Einstein summation convention.
- Tensor Types: Contravariant, covariant, and mixed tensors; Kronecker delta property.
- Tensor Algebra: Contraction, inner product, outer product, and quotient law.
- Metric Tensor: Riemannian space, line elements, and reciprocal tensors.
- Christoffel Symbols: Symbols of the first and second kind, along with covariant differentiation of tensors.

3. Linear Vector Space

This algebraic section establishes the formal mathematical rigorous framework necessary for understanding modern quantum mechanics.

Vector Space Axioms: Definition of a linear vector space over fields of real or complex numbers.
Linear Dependence: Linear combinations, linear dependence, and independence of vectors.
Basis and Dimension: Defining a basis set, span, and the dimension of vector spaces.
Inner Product Spaces: Definition of inner products, norm/length of a vector, and the Cauchy-Schwarz inequality.
Orthogonality: Orthonormal basis systems and the Gram-Schmidt orthogonalization process.
Linear Operators & Matrices: Linear transformations, eigenvalues, eigenvectors, and matrix diagonalization.