Derivatives of Vectors and Tensors

Synonyms

Del operations; Nabla operations; Vector differential operators

Definitions

A tensor field is a tensor-valued function of position in space. Hence a scalar field is a mapping that assigns a scalar to each point of some domain in $\mathbb R^3$; a vector field assigns a vector to each point of such a domain, etc. The basic notion of derivative can be applied to a tensor field of any order, yielding information about its spatial rates of change within the domain of interest.

Derivatives of Vector Fields in Curvilinear Coordinates

In elementary multivariable calculus, the differential operations for vector functions are presented in Cartesian coordinates. Hence they reduce to mere differentiation of the vector components, as the derivatives of the basis vectors vanish. In continuum mechanics, however, curvilinear coordinates (denoted here by q ⁱ) are indispensable. Differentiation of vector functions given in curvilinear coordinates entails differentiation of the basis vectors,...