Tensor

In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can be defined in several different ways, often a scalar, tangent vector at a point, a cotangent vector (dual vector) at a point, or a multi-linear map from vector spaces to a resulting vector space. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors. While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system.Definition. A tensor of type (p, q) is an assignment of a multidimensional arrayI 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. In mathematics, a tensor is an algebraic object related to a vector space and its dual space that can be defined in several different ways, often a scalar, tangent vector at a point, a cotangent vector (dual vector) at a point, or a multi-linear map from vector spaces to a resulting vector space. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors. While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system. An elementary example of mapping, describable as a tensor, is the dot product, which maps two vectors to a scalar. A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The cross product, where two vectors are mapped to a third one, is strictly speaking not a tensor, because it changes its sign under those transformations that change the orientation of the coordinate system. The totally anti-symmetric symbol ε i j k {displaystyle varepsilon _{ijk}} nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems. Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing ε i j k {displaystyle varepsilon _{ijk}} not being a tensor, for the sign change under transformations changing the orientation. Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. The order of a tensor is the sum of these two numbers. The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. For example in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, equal to the stress tensor, taking one vector and returning another 1 + 1 = 2. The ε i j k {displaystyle varepsilon _{ijk}} -symbol, mapping two vectors to one vector, would have order 2 + 1 = 3. The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order 2, which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this. Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ... ) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are simply called 'tensors'. Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.