Space, and such tensors are known as Cartesian While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean Note that the positions of the slots in which contravariant and covariant indicesĪre placed are significant so, for example, is distinct from. In addition, a tensor with rank may be of mixed type, consisting of so-called "contravariant" (upper) indices The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor, ,, etc., may have an arbitrary number of indices. ![]() Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity. Have exactly two indices) to an arbitrary number of indices. (that have exactly one index), and matrices (that Tensors are generalizations of scalars (that have no indices), vectors (with the notable exception of the contracted Kroneckerĭelta). However, the dimension of the space is largely irrelevant in most tensor equations Of a tensor ranges over the number of dimensions of space. Return the vector field module on which self acts as a tensor.Tensor in -dimensional space is a mathematicalĬomponents and obeys certain transformation rules. coord ( c_uv ) # to check the above expression (5, -1) base_module ( ) # parent () Free module of type-(1,1) tensors on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: ap. at ( p ) ap Type-(1,1) tensor a on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: ap. point (( 2, 3 ), chart = c_xy, name = 'p' ) sage: ap = a. add_comp_by_continuation ( eV, W, chart = c_uv ) sage: a. ![]() Non-negative integers \(k\) and \(l\) and a differentiable map Manifold \(U\) with values on a differentiable manifold \(M\), via aĭifferentiable map \(\Phi: U \rightarrow M\). Tensor field along a differentiable manifold.Īn instance of this class is a tensor field along a differentiable TensorField ( vector_field_module : VectorFieldModule, tensor_type : TensorType, name : Optional = None, latex_name : Optional = None, sym = None, antisym = None, parent = None ) # Michael Jung (2019): improve treatment of the zero element add methodĮric Gourgoulhon (2020): add method TensorField.apply_map()Ĭlass. ![]() Various derived classes of TensorField are devoted to specific tensorįields (rank-1 contravariant tensor fields)įor fields of tangent-space automorphismsįorms (fully antisymmetric covariant tensor fields)įor multivector fields (fully antisymmetric contravariant tensor fields)Įric Gourgoulhon, Michal Bejger (2013-2015) : initial versionĮric Gourgoulhon (2018): operators divergence, Laplacian and d’Alembertian įlorentin Jaffredo (2018) : series expansion with respect to a given Is devoted to tensor fields with values on parallelizable manifolds. The class TensorField implements tensor fields on differentiable Toggle table of contents sidebar Tensor Fields #
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