Tensors for Physics

Chapter 7

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 intro-

duced 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 dis-

cussed. An application involves the pressure tensor in the local conservation law for

linear momentum. Further applications are the Maxwell equations of electrodynam-

ics 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.

A functionf = f(r)which determines a number at any space pointris called
afield. In three dimensional space (3D),fis a function of the three components
r 1 ,r 2 ,r 3 of the position vector. Such a function, in turn, can be a scalar, a component
of a vector or of a tensor. Depending on the rank of the tensor, one talks of

1. Scalar fields, examples are:
   the potential energyΦ=Φ(r)or the electrostatic potentialφ=φ(r).


2. Vector fields, like
   the forceF=F(r), the electric fieldE=E(r),or
   the flow fieldv=v(r)of hydrodynamics.


3. Tensor fields, an example for a second rank tensor field is
   the pressure tensor or the stress tensor.

© Springer International Publishing Switzerland 2015

S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,

DOI 10.1007/978-3-319-12787-3_7

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