In general, scalar elds are referred to as tensor elds of rank or order zero whereas vector elds are called tensor elds of rank or order one. This work covers all the basic topics of tensor analysis in a lucid and clear language and is aimed at both the undergraduate and postgraduate in Civil, Mechanical and Aerospace Engineering and in Engineering Physics. When these numbers obey certain transformation laws they become examples of tensor elds. Introduction / Some Preliminaries: Introduction / Systems of Different Orders / Sumination Convention / Kronecker Symbols / Some Results of Determinant / Differentiation of a Determinant / Linear Equations, Cramer's Rule / Examples / Exercises / Tensor Algebra: Introduction / n-dimensional Space / Transformation of Coordinates in Sn / Invariants / Vectors / Covariant Vectors / Contravariant Vectors / Tensors of Second Order / Contravariant Tensors of Order Two / Convariant Tensors of Order Two / Mixed Tensors of Order Two / Mixed Tensors of Type (p, q) / Zero Tensor / Tensor Field / Algebra of Tensors / Equality of Two Tensors / Symmetric and Skew-symmetric Tensors / Symmetric Tensors /Skew-symmetric Tensors / Outer Multiplications and Contraction / Outer Multiplications / Contraction / Quotient Law of Tensors / Quotient Law of Tensors of First Order and Type (0,1) / Quotient Law for Tensors of First Order and Type (1,0) / Quotient Law for Convariant Tensors of Second Order or of Type (0,2) / Quotient Law for Contravariant Tensors of Second Order or of Type (2,0) / Quotient Law in General Form / Reciprocal Tensor of a Tensor / Relative Tensor / Cross Product or Vector Product of Two Vectors / Examples / Exercises / Tensor Calculus: Introduction / Riemannian Space / Riemannian Metric / reciprocal or Conjugate Tensor of the Fundamental Metric Tensor gij / Associated Tensors, Lowering and Raising Indices / Magnitude or Length of a Vector / Unit Vector / Null Vector / Angle Between Two Non-null Vectors / Orthogonal Vectors / Christoffel Symbols and their Properties / Christoffel Symbols of First Kind / Christoffel Symbols of Second Kind / Properties of Christoffel Symbols / Law of Transformation of Christoffel Symbols of First Kind / Law of Transformation of Christoffel Symbols of Second Kind / Covariant Differentiation of Tensors / Covariant Differentiation of a Covariant Vector / Covariant Differentiation of a Contravariant Vector / Covariant Differentiation of Tensors of Type (0,2) / Covariant Derivative of a Tensor of Type (2,0) / Covariant Derivative of a Mixed tensor of type (1,1) / Covariant Derivate of a Mixed Tensor of Type (p, q) / Ricci's Theorem / Gradient of a Scalar / Divergence of a Contravariant Vector / Divergence of a Covariant Vector / Conservative Vector / Divergence of a Contravariant Tensor of Order Two / Divergence of a Mixed Tensor of Type (1,1) / Laplacian of an Invariatn / Curl of a Covariant Vector / Riemann-Christoffel Curvature Tensor / Definition / Properties of Tiemann-Christoffel Curvature Tensor / Rcci Tensor / Scalar Curvature / Einstein Space / Einstein Space / Einstein Tensor / Intrinsic Differentiation / Geodesics, Riemannian Coordinates and Geodesic Coordinates: Calculus of Variations / Families of Curves / Euler's Conditions / Geodesics / Riemannian and Geodesic Coordinates / History of Tensor Calculus.
_x f(t)\,dt=f(c).Bibliography Includes bibliographical references (p. Specifically to the problem raised in Example 2.1, we shall show coordinate-free universal algebraic formulation of the gradient vector.
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