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“Mathematics is a language.” Thus was the response of the great American scientist J. Willard Gibbs when asked at a Yale faculty meeting whether mathematics should really be as important a part of the undergraduate curriculum as classical languages.
Tensor calculus is a specific language within the general language of mathematics. It is used to express the concepts of multivariable calculus and its applications in disciplines as diverse as linear algebra, differential geometry, calculus of variations, continuum mechanics, and, perhaps tensors’ most popular application, general relativity. Albert Einstein was an early proponent of tensor analysis and made a valuable contribution to the subject in the form of the Einstein summation convention. Furthermore, he lent the newly invented technique much clout and contributed greatly to its rapid adoption. In a letter to Tullio Levi-Civita, a co-inventor of tensor calculus, Einstein expressed his admiration for the subject in the following words: “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.”
Tensor calculus is not the only language for multivariable calculus and its applications. A popular alternative to tensors is the so-called modern language of differential geometry. Both languages aim at a geometric description independent of coordinate systems. Yet, the two languages are quite different and each offers its own set of relative strengths and weaknesses.
