In contrast, for standard calculus, the gradient vector formula is dependent on the coordinate system in use (example: Cartesian gradient vector formula vs. In fact in conventional math syntax we make use of covariant indexes when dealing with Cartesian coordinate systems ( x 1, x 2, x 3 ) (x_ Tensor notation makes use of upper and lower indexes on objects that are used to label a variable object as covariant (lower index), contravariant (upper index), or mixed covariant and contravariant (having both upper and lower indexes). So to handle this, you need the Ricci calculus. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily.
In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. A description of aphysical process cannot dependon the coordinate orientation or origin, and this principlecan be expanded to look for othermathematical symmetries. They are allowed to undergo transformation. The concept of a tensor is derived from considering the properties of a function under atransformation of the coordinate system. Vectors, Tensors and the Basic Equations of Fluid. In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. calculus, as well as a brief glimpse into the subjects manifold applications. Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus: Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
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. in spacetime).ĭeveloped by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields ( tensors that may vary over a manifold, e.g. It has been suggested that this article be merged into Ricci calculus.
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