  We derive a number of local identities of arbitrary rank involving Jacobi
elliptic functions and use them to obtain several new results. First, we
present an alternative, simpler derivation of the cyclic identities discovered
by us recently, along with an extension to several new cyclic identities of
arbitrary rank. Second, we obtain a generalization to cyclic identities in
which successive terms have a multiplicative phase factor exp(2i\pi/s), where s
is any integer. Third, we systematize the local identities by deriving four
local ``master identities'' analogous to the master identities for the cyclic
sums discussed by us previously. Fourth, we point out that many of the local
identities can be thought of as exact discretizations of standard nonlinear
differential equations satisfied by the Jacobian elliptic functions. Finally,
we obtain explicit answers for a number of definite integrals and simpler forms
for several indefinite integrals involving Jacobi elliptic functions.
