Fields, Spatial Differential Operators

Abstract

This chapter is devoted to the spatial differentiation of fields which are tensors of various ranks and to the properties of spatial differential operators. Firstly, scalar fields like potential functions are considered. The nabla operator is introduced and applications of gradient fields are discussed, e.g. force fields in Newton’s equation of motion. Secondly, the differential change of vector fields is analyzed, the divergence and the curl or rotation of vector fields are defined. Special types of vector fields are studied: vorticity-free fields as derivatives of scalar potentials and divergence-free fields as derivatives of vector potentials. The Laplace operator, the Laplace and Poisson equations are introduced. The conventional classification of vector fields is listed. Thirdly, tensor fields are considered. A graphical representation of symmetric second rank tensors is given. Spatial derivatives of tensor fields are discussed. An application involves the pressure tensor in the local conservation law for linear momentum. Further applications are the Maxwell equations of electrodynamics in differential form. This chapter is concluded by rules for the nabla and Laplace operators, their decomposition into radial and angular parts, with applications to the orbital angular momentum and kinetic energy operators of Wave Mechanics.