Extensorial

Extensorial refers to a mathematical concept often encountered in differential geometry and tensor calculus. It describes a particular type of tensor, specifically a tensor that is constructed from a vector space and its dual space. More precisely, an extensorial is a multilinear map from a product of copies of the vector space and its dual space to the underlying field of scalars. This means it takes multiple vector arguments and multiple covector arguments and produces a single scalar output, behaving linearly with respect to each argument independently. The order of the tensor, its rank, is determined by the number of vector and covector arguments it accepts. For instance, a tensor of type (p, q) is an extensorial that takes p covectors and q vectors as input. Extensorials are fundamental objects used to express geometric quantities such as curvature, stress, and strain in a coordinate-independent manner. They provide a powerful framework for describing physical laws and geometric properties that are invariant under coordinate transformations. The manipulation and transformation of extensorials under changes of basis are central to tensor analysis.