Brannon R.F. Functional and Structured Tensor Analysis for Engineers

A casual (intuition-based) introduction to vector and tensor analysis with reviews of popular notations used in contemporary materials modeling. – University of New Mexico, 2003. – 323 pp. Draft.Elementary vector and tensor analysis concepts are reviewed in a manner that proves useful for higher-order tensor analysis of anisotropic media. In addition to reviewing basic matrix and vector analysis, the concept of a tensor is covered by reviewing and contrasting numerous different definition one might see in the literature for the term “tensor.” Basic vector and tensor operations are provided, as well as some lesser-known operations that are useful in materials modeling. Considerable space is devoted to “philosophical” discussions about relative merits of the many (often conflicting) tensor notation systems in popular use.Terminology from functional analysis
Matrix Analysis (and some matrix calculus)
Vector/tensor notation
Simple vector operations and properties
Projections
Tensors
Tensor operations
Tensor analysis primitives
Projectors in tensor notation
More Tensor primitives
Tensor operations
Coordinate/basis transformations
Tensor (and Tensor function) invariance
Scalar invariants and spectral analysis
Geometrical entities
Polar decomposition
Material symmetry
Abstract vector/tensor algebra
Vector and Tensor Visualization
Vector/tensor differential calculus
Vector/tensor Integral calculus
Closing remarks
Solved problems