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In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields.
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism , general relativity , quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus: