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• Ricci_calculus abstract "In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887–96, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.A component of a tensor is a real number which is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their basis elements. Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows the most efficient expressions of such tensor fields and operations. While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays.A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. The number of indices equals the order of the tensor.For compactness and convenience, the notational convention implies certain things, notably that of summation over indices repeated within a term and of universal quantification over free indices (those not so summed). Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.".
• Ricci_calculus wikiPageExternalLink theoryofrelativi029229mbp.
• Ricci_calculus wikiPageExternalLink books?as_isbn=140201015X.
• Ricci_calculus wikiPageID "35456546".
• Ricci_calculus wikiPageLength "36374".
• Ricci_calculus wikiPageOutDegree "107".
• Ricci_calculus wikiPageRevisionID "667255580".
• Ricci_calculus wikiPageWikiLink Abstract_index_notation.
• Ricci_calculus wikiPageWikiLink Active_and_passive_transformation.
• Ricci_calculus wikiPageWikiLink Antisymmetric_tensor.
• Ricci_calculus wikiPageWikiLink Array_data_type.
• Ricci_calculus wikiPageWikiLink Base_manifold.
• Ricci_calculus wikiPageWikiLink Basis_(linear_algebra).
• Ricci_calculus wikiPageWikiLink Bracket.
• Ricci_calculus wikiPageWikiLink Cartesian_coordinate_system.
• Ricci_calculus wikiPageWikiLink Cartesian_coordinates.
• Ricci_calculus wikiPageWikiLink Category:Differential_geometry.
• Ricci_calculus wikiPageWikiLink Category:Tensors.
• Ricci_calculus wikiPageWikiLink Christoffel_symbol.
• Ricci_calculus wikiPageWikiLink Christoffel_symbols.
• Ricci_calculus wikiPageWikiLink Comma.
• Ricci_calculus wikiPageWikiLink Commutator.
• Ricci_calculus wikiPageWikiLink Connection_(mathematics).
• Ricci_calculus wikiPageWikiLink Coordinate_system.
• Ricci_calculus wikiPageWikiLink Coordinate_vector.
• Ricci_calculus wikiPageWikiLink Covariance_and_contravariance_of_vectors.
• Ricci_calculus wikiPageWikiLink Covariant_derivative.
• Ricci_calculus wikiPageWikiLink Covector.
• Ricci_calculus wikiPageWikiLink Diacritic.
• Ricci_calculus wikiPageWikiLink Differential_form.
• Ricci_calculus wikiPageWikiLink Differential_geometry.
• Ricci_calculus wikiPageWikiLink Differential_structure.
• Ricci_calculus wikiPageWikiLink Dimension.
• Ricci_calculus wikiPageWikiLink Dimensional_analysis.
• Ricci_calculus wikiPageWikiLink Directional_derivative.
• Ricci_calculus wikiPageWikiLink Distributive_property.
• Ricci_calculus wikiPageWikiLink Einstein_notation.
• Ricci_calculus wikiPageWikiLink Einstein_summation_convention.
• Ricci_calculus wikiPageWikiLink Electromagnetic_tensor.
• Ricci_calculus wikiPageWikiLink Euclidean_space.
• Ricci_calculus wikiPageWikiLink Exterior_algebra.
• Ricci_calculus wikiPageWikiLink Exterior_calculus.
• Ricci_calculus wikiPageWikiLink Exterior_derivative.
• Ricci_calculus wikiPageWikiLink Faradays_law_of_induction.
• Ricci_calculus wikiPageWikiLink Fiber_bundle.
• Ricci_calculus wikiPageWikiLink Four-dimensional_space.
• Ricci_calculus wikiPageWikiLink Frame_of_reference.
• Ricci_calculus wikiPageWikiLink Gausss_law_for_magnetism.
• Ricci_calculus wikiPageWikiLink General_relativity.
• Ricci_calculus wikiPageWikiLink Generalized_Kronecker_delta.
• Ricci_calculus wikiPageWikiLink Greek_alphabet.
• Ricci_calculus wikiPageWikiLink Gregorio_Ricci-Curbastro.
• Ricci_calculus wikiPageWikiLink Hodge_dual.
• Ricci_calculus wikiPageWikiLink Holonomic_basis.
• Ricci_calculus wikiPageWikiLink Identity_function.
• Ricci_calculus wikiPageWikiLink Identity_mapping.
• Ricci_calculus wikiPageWikiLink Identity_matrix.
• Ricci_calculus wikiPageWikiLink If_and_only_if.
• Ricci_calculus wikiPageWikiLink Inverse_matrix.
• Ricci_calculus wikiPageWikiLink Invertible_matrix.
• Ricci_calculus wikiPageWikiLink Jan_Arnoldus_Schouten.
• Ricci_calculus wikiPageWikiLink Kronecker_delta.
• Ricci_calculus wikiPageWikiLink Latin_alphabet.
• Ricci_calculus wikiPageWikiLink Levi-Civita_symbol.
• Ricci_calculus wikiPageWikiLink Lie_derivative.
• Ricci_calculus wikiPageWikiLink Line_element.
• Ricci_calculus wikiPageWikiLink Linear_form.
• Ricci_calculus wikiPageWikiLink Lorentz_transformation.
• Ricci_calculus wikiPageWikiLink Mathematics.
• Ricci_calculus wikiPageWikiLink Metric_tensor.
• Ricci_calculus wikiPageWikiLink Mixed_tensor.
• Ricci_calculus wikiPageWikiLink Monotonic_function.
• Ricci_calculus wikiPageWikiLink Multi-index_notation.
• Ricci_calculus wikiPageWikiLink Multidimensional_array.
• Ricci_calculus wikiPageWikiLink Parameterization.
• Ricci_calculus wikiPageWikiLink Parametrization.
• Ricci_calculus wikiPageWikiLink Parity_of_a_permutation.
• Ricci_calculus wikiPageWikiLink Partial_derivative.
• Ricci_calculus wikiPageWikiLink Passive_transformation.
• Ricci_calculus wikiPageWikiLink Penrose_graphical_notation.
• Ricci_calculus wikiPageWikiLink Permutation.
• Ricci_calculus wikiPageWikiLink Product_rule.
• Ricci_calculus wikiPageWikiLink Raising_and_lowering_indices.
• Ricci_calculus wikiPageWikiLink Real_number.
• Ricci_calculus wikiPageWikiLink Regge_calculus.
• Ricci_calculus wikiPageWikiLink Relative_tensor.
• Ricci_calculus wikiPageWikiLink Ricci_decomposition.
• Ricci_calculus wikiPageWikiLink Riemann_curvature_tensor.
• Ricci_calculus wikiPageWikiLink Semicolon.
• Ricci_calculus wikiPageWikiLink Signature_(permutation).
• Ricci_calculus wikiPageWikiLink Slash_(punctuation).
• Ricci_calculus wikiPageWikiLink Smooth_function.
• Ricci_calculus wikiPageWikiLink Smoothness.
• Ricci_calculus wikiPageWikiLink Space-like.
• Ricci_calculus wikiPageWikiLink Spacetime.
• Ricci_calculus wikiPageWikiLink Spinor.
• Ricci_calculus wikiPageWikiLink Symmetric_tensor.
• Ricci_calculus wikiPageWikiLink Tangent_bundle.
• Ricci_calculus wikiPageWikiLink Tensor.
• Ricci_calculus wikiPageWikiLink Tensor_(intrinsic_definition).

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