Fundamental Tensor Operations to Know for Tensor Analysis

Fundamental tensor operations form the backbone of tensor analysis, enabling us to manipulate and understand tensors effectively. These operations, including addition, multiplication, contraction, and transformation, are essential for applications in physics, engineering, and data science.

1. Tensor addition and subtraction

   • Tensors of the same type and order can be added or subtracted element-wise.
   • The result of addition or subtraction is another tensor of the same order.
   • This operation is commutative and associative, similar to vector addition.

2. Tensor multiplication (outer product)

   • The outer product combines two tensors to create a new tensor of higher order.
   • For tensors A (of order m) and B (of order n), the outer product results in a tensor of order m+n.
   • The outer product is not commutative; the order of the tensors matters.

3. Tensor contraction

   • Contraction reduces the order of a tensor by summing over one or more pairs of indices.
   • This operation is crucial for simplifying tensor expressions and extracting scalar quantities.
   • Contraction can be thought of as a generalization of the dot product for higher-order tensors.

4. Tensor inner product

   • The inner product of two tensors produces a scalar value.
   • It involves summing the products of corresponding components of the tensors.
   • The inner product is a specific case of contraction, typically applied to tensors of the same order.

5. Tensor transpose

   • The transpose of a tensor rearranges its indices, effectively flipping it across its diagonal.
   • For a second-order tensor, transposition switches its rows and columns.
   • Transposition is essential for defining symmetric and antisymmetric tensors.

6. Tensor symmetrization and antisymmetrization

   • Symmetrization creates a new tensor that is invariant under the exchange of specified indices.
   • Antisymmetrization produces a tensor that changes sign when indices are exchanged.
   • These operations are important in physics, particularly in the study of angular momentum and other properties.

7. Tensor differentiation

   • Tensor differentiation extends the concept of differentiation to tensor fields.
   • It involves computing the derivative of tensor components with respect to a variable, often a coordinate.
   • This operation is essential in formulating physical laws in continuum mechanics and general relativity.

8. Tensor integration

   • Tensor integration involves integrating tensor fields over a specified domain.
   • It generalizes the concept of integration to account for the multi-dimensional nature of tensors.
   • This operation is crucial in applications such as fluid dynamics and electromagnetism.

9. Tensor decomposition

   • Tensor decomposition breaks down a tensor into simpler, constituent tensors.
   • Common methods include CANDECOMP/PARAFAC and Tucker decomposition.
   • Decomposition aids in data analysis, compression, and understanding the structure of tensors.

10. Tensor transformation under coordinate changes

    • Tensors transform according to specific rules when changing coordinate systems.
    • The transformation involves applying a linear transformation to the tensor components based on the Jacobian matrix.
    • Understanding these transformations is vital for ensuring the consistency of physical laws across different reference frames.