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I would like to understand how the concept of tensor emerged and then following the success of General Relativity gradually become mature as tensor analysis. The axiomatic approach isn't helping understanding the motivation nor developing the geometric intuition.

One way could be learn from the masters who created the tensor analysis. Please suggest some accessible references. I wonder whether there is any recent book which present the tensor analysis as it is developed in the history; and then tie it up with the modern textbook presentation of the tensor analysis.

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Wikipedia covers the history of tensors and cites some primary sources; I'll try to add a few additional sources here. I'm not linking to Wikipedia as I don't have enough reputation to post more than 8 links.

Gregorio Ricci-Curbastro is considered a creator of tensor analysis, and together with his student Tullio Levi-Civita he wrote Méthodes de calcul différentiel absolu et leurs applications (1901). This paper is frequently cited as having introduced tensor analysis to a wider audience. Robert Hermann produced an English-language translation, though there are some sections left untranslated by Hermann.

More details about this paper are found in the "Tensor Analysis" section of Toward a scientific and personal biography of Tullio Levi-Civita (1873–1941):

In 1899 Felix Klein (1849–1925) met Levi-Civita in Padua and asked him to publish in his journal, Mathematische Annalen, an organic and systematic account of tensor calculus.

More details on how Ricci developed his theory of absolute differential calculus (i.e. tensor analysis) can be found in Luca Dell'Aglio's On the genesis of the concept of covariant differentiation (1996).

More details on the Italian scene in which Ricci and Levi-Civita worked can be found in Judith Goodstein's The Italian Mathematicians of Relativity (1982).

For more on how Einstein picked these ideas up from Levi-Civita while developing his theory of general relativity see Lost in the Tensors: Einstein's Struggles with Covariance Principles, 1912-1916 (1978), How Einstein Found His Field Equations: 1912-1915 (1984), or Einstein the Stubborn: Correspondence between Einstein and Levi-Civita (2012).

Through the examination of these sources I'm sure you can turn up additional primary texts to read.

Now to your question about a textbook which tries to motivate a modern presentation of tensor analysis with historical context: I don't know that such a book exists, unfortunately.

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