Mathematical Methods for Physicists: A concise introduction - Site Map

TENSOR ANALYSIS
Proof: In Gauss' theorem (1.78), let A ˆ B C where C is a constant vector. We
then have
Z
Z
r…B C†dV ˆ …B C†^nda:
V
S
Since r…B C† ˆC …rB† and …B C†^n ˆ B …C ^n† ˆ…C ^n†B ˆ
C …^n B†;
Z
Z
C …rB†dV ˆ C …^n B†da:
V
S
Taking C outside the integrals
Z
Z
C …r B†dV ˆ C …^n B†da
V
S
and since C is an arbitrary constant vector, we have
Z
Z
rBdV ˆ ^n Bda:
V
S
Tensor analysis
Tensors are a natural generalization of vectors. The beginnings of tensor analysis
can be traced back more than a century to Gauss' works on curved surfaces.
Today tensor analysis ®nds applications in theoretical physics (for example, general
theory of relativity, mechanics, and electromagnetic theory) and to certain
areas of engineering (for example, aerodynamics and ¯uid mechanics). The general
theory of relativity uses tensor calculus of curved space-time, and engineers
mainly use tensor calculus of Euclidean space. Only general tensors are
considered in this section. The general de®nition of a tensor is given, followed
by a concise discussion of tensor algebra and tensor calculus (covariant di€erentiation).
Tensors are de®ned by means of their properties of transformation under
coordinate transformation. Let us consider the transformation from one coordinate
system …x 1 ; x 2 ; ...; x N † to another …x 01 ; x 02 ; ...; x 0N † in an N-dimensional
space V N . Note that in writing x , the index is a superscript and should not
be mistaken for an exponent. In three-dimensional space we use subscripts. We
now use superscripts in order that we may maintain a `balancing' of the indices in
all the general equations. The meaning of `balancing' will become clear a little
later. When we transform the coordinates, their di€erentials transform according
to the relation
dx ˆ @x
@x 0 dx 0 :
…1:94†
47