"Tensor Analysis on Manifolds " by Bishop and Goldberg; it's friendly, inexpensive, light on prereqs, rigorus, and covers a decent ammount of topics for an introduction ( in my opinion ).
22.
In mathematics, "'tensor calculus "'or "'tensor analysis "'is an extension of vector calculus to tensor fields ( tensors that may vary over a manifold, e . g . in spacetime ).
23.
The works of physicists such as James Clerk Maxwell, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of atlas is due to Hassler Whitney.
24.
In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold.
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This culminated in 1910 with the publication of Woldemar Voigt's " Lehrbuch der Kristallphysik " ( " Textbook on Crystal Physics " ), which described the 20 natural crystal classes capable of piezoelectricity, and rigorously defined the piezoelectric constants using tensor analysis.
26.
The use of mixed contributions from gradients in the three primary orthogonal axes in order to generate an infinite number of differently oriented gradients for tensor analysis was also identified in 1992 as the basis for accomplishing tensor descriptions of water diffusion in MRI voxels.
27.
At UIUC, his doctoral students included future UIUC colleague Stephanie B . Alexander . translated into Russian 1967 and reprinted 2001 ) and " Tensor Analysis on Manifolds " ( with Samuel I . Goldberg, Macmillan, 1968, reprinted by Dover Books on Mathematics, 1980 ).
28.
Some decades later the rather abstract view coming from category theory was tied up with the approach that had been developed in the 1930s by Hermann Weyl ( by working through general relativity via abstract tensor analysis, and additionally in his book " The Classical Groups " ).
29.
When doing linear algebra or tensor analysis, if you write an expression in terms of its components in some basis ( or local coordinates ), but the total expression is invariant under coordinate changes ( all the component indices are contracted ?la Einstein ), then there should be a coordinate-free way of writing the expression.
