How is tensor algebra used in tensor analysis?

Is there a contraction of a pair of tensors?

Contraction of a pair of tensors. The tensor product is a new tensor, which, if it has at least one covariant and one contravariant index, can be contracted. The case where T is a vector and U is a dual vector is exactly the core operation introduced first in this article.

Can a tensor contract if its base vectors are dotted?

This tensor does not contract; if its base vectors are dotted, the result is the contravariant metric tensor , whose rank is 2. As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general.

How is the contraction of a vector denoted?

Contraction in index notation. In tensor index notation, the basic contraction of a vector and a dual vector is denoted by. which is shorthand for the explicit coordinate summation. (where v i are the components of v in a particular basis and f i are the components of f in the corresponding dual basis).

When to use metric contraction and tensor contraction?

One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known as metric contraction. Contraction is often applied to tensor fields over spaces (e.g. Euclidean space, manifolds, or schemes ).

How is tensor algebra used in tensor analysis?

Tensor Algebra Tensor Algebra Table of contents Multilinear functions and tensors Tensor products Basis representation Change of basis Contraction Generalized dot product of tensors Volume forms Linear Maps – II Linearization of Nonlinear Maps Euclidean Tensor Analysis Curvilinear Coordinates

How to contract a network with n > 2 tensors?

Broadly speaking, there are two approaches that could be taken to contract a network containing N>2 tensors: (i) in a single step as a direct summation over all internal indices of the network or (ii) as a sequence of N-1 binary contractions.

Can a contorder be omitted from a tensor contraction?

Note that ‘ContOrder’ is an optional input that can be omitted if desired, in which case ‘ncon’ will contract in ascending order of index labels. If a pair of tensors is connected via multiple indices then ‘ncon’ will perform the contraction as a single multiplication (as opposed to contracting each index sequentially).

How are the contractions of a tensor network evaluated?

Given a tensor network composed of N tensors, there are two distinct steps needed to contract the network efficiently: evaluate each of the binary contractions in turn as a matrix multiplication by taking the proper tensor permutes and reshapes.

How to calculate the optimal sequence of tensor contractions?

Given a tensor network composed of N tensors, there are two distinct steps needed to contract the network efficiently: 1 determine the optimal sequence of the (N-1) binary tensor contractions, 2 evaluate each of the binary contractions in turn as a matrix multiplication by taking the proper tensor permutes and… More

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