The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor. The contemporary usage was brought in by Woldemar Voigt in 1898.
Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892. It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).
In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot. —Albert Einstein, The Italian Mathematicians of Relativity
Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensors used in mathematics.
From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem). Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field, but the theory is then certainly less geometric, and computations more technical and less algorithmic. Tensors are generalized within category theory by means of the concept of monoidal category, from the 1960s.
Read more about this topic: Tensor
Famous quotes containing the word history:
“In the history of the United States, there is no continuity at all. You can cut through it anywhere and nothing on this side of the cut has anything to do with anything on the other side.”
—Henry Brooks Adams (18381918)
“The thing that struck me forcefully was the feeling of great age about the place. Standing on that old parade ground, which is now a cricket field, I could feel the dead generations crowding me. Here was the oldest settlement of freedmen in the Western world, no doubt. Men who had thrown off the bands of slavery by their own courage and ingenuity. The courage and daring of the Maroons strike like a purple beam across the history of Jamaica.”
—Zora Neale Hurston (18911960)
“In history the great moment is, when the savage is just ceasing to be a savage, with all his hairy Pelasgic strength directed on his opening sense of beauty;and you have Pericles and Phidias,and not yet passed over into the Corinthian civility. Everything good in nature and in the world is in that moment of transition, when the swarthy juices still flow plentifully from nature, but their astrigency or acridity is got out by ethics and humanity.”
—Ralph Waldo Emerson (18031882)