  The structures and symmetries of single-walled carbon nanotubes (SWNTs) are
introduced in detail. The physical properties of SWNTs induced by their
symmetries can be described by tensors in mathematical point of view. It is
found that there are 2, 4, and 5 different parameters in the second, third, and
fourth rank tensors representing electronic conductivity (or static
polarizability), the second order nonlinear polarizability, and elastic
constants of SWNTs, respectively. The values of elastic constants obtained from
tight-binding method imply that SWNTs might be very weakly anisotropic in
mechanical properties. The further study on the mechanical properties shows
that the elastic shell theory in the macroscopic scale can be applied to carbon
nanotubes (CNTs) in the mesoscopic scale, as a result, SWNTs can be regarded as
an isotropic material with Poisson ratio, effective thickness, and Young's
modulus being $\nu=0.34$, $h=0.75$\AA, $Y=4.70$TPa, respectively, while the
Young's moduli of multi-walled carbon nanotubes (MWNTs) are apparent functions
of the number of layers, $N$, varying from 4.70TPa to 1.04TPa for N=1 to
$\infty$. Based on the chirality of CNTs, it is predicted that a new kind of
molecular motor driven by alternating voltage can be constructed from double
walled carbon nanotubes (DWNTs).
