differential geometry and tensor analysis

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. They are allowed to undergo transformation. 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. 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. 'Tensor.It has been suggested that this article be merged into Ricci calculus. Sokolnikoff,Īnalysis: Theory and Applications to Geometry and Mechanics of Continua, 2nd ed. Simmonds, J. G.īrief on Tensor Analysis, 2nd ed. Riemann'sche Geometrie und Tensoranalysis.īerlin: Deutscher Verlag der Wissenschaften, 1959. Parker,Ī System for Doing Tensor Analysis by Computer. Tensors,ĭifferential Forms, and Variational Principles. Introduction to Tensor Calculus, Relativity, and Cosmology, 3rd ed. Borisenko, A. I.Īnd Tensor Analysis with Applications. Tensors, and the Basic Equations of Fluid Mechanics. Orlando, FL: Academic Press, pp. 118-167, Introduction to Linear Algebra and Tensors. Tensor Analysis, and Applications, 2nd ed. Hemisphere moment of inertia tensor of solid These can be achieved through multiplication by a so-called metricĬircle area moment of inertia tensor of enclosed laminaĬonical frustum moment of inertia tensor of solid Lowering and index raising as special cases. Of tensor indices to produce identities or to simplify expressions is known as index gymnastics, which includes index Kronecker delta) or by tensor operators (suchĪs the covariant derivative). Tensors may be operated on by other tensors (such as metric tensors, the permutation tensor, or the , which would be written in tensor notation.

Objects that transform like zeroth-rank tensors are called scalars, those that transform like first-rank tensors are called vectors,Īnd those that transform like second-rank tensors are called matrices. Space, and such tensors are known as Cartesian While the distinction between covariant and contravariant indices must be made for general tensors, the two are equivalent for tensors in three-dimensional Euclidean Note that the positions of the slots in which contravariant and covariant indicesĪre placed are significant so, for example, is distinct from. In addition, a tensor with rank may be of mixed type, consisting of so-called 'contravariant' (upper) indices The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor, ,, etc., may have an arbitrary number of indices. Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity. Have exactly two indices) to an arbitrary number of indices.

(that have exactly one index), and matrices (that Tensors are generalizations of scalars (that have no indices), vectors (with the notable exception of the contracted Kroneckerĭelta). However, the dimension of the space is largely irrelevant in most tensor equations Of a tensor ranges over the number of dimensions of space. Tensor in -dimensional space is a mathematicalĬomponents and obeys certain transformation rules.

Differential geometry and tensor analysis