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integral (sub)manifold (of a distribution of sub- into In mathematics, sometimes the word is
spaces of rank n) A submanifold N ⊂ M used to identify a one-to-one mapping, some-
such that T N ⊂ for all x ∈ N. Usually N times simply a mapping of a set X which trans-
x x
is required to be of dimension n = rank( ) forms points of X into points of another set Y,
2
e.g., y = x is a mapping of the real numbers
integrable system A Hamiltonian system into the real numbers, or onto the nonnegative
X H ona2n-dimensional symplectic manifold real numbers. See also onto, bijection, injection,
(Mω) such that there exist n linearly indepen- and surjection.
dent functions H = F ,...,F in involution,
1 n
i.e., {F ,F }= 0,i, j = 1,...,n and dF (m) intramolecular (1) Descriptive of any pro-
i
i
j
are linearly independent at each point m ∈ M. cess that involves a transfer (of atoms, groups,
electrons, etc.) or interactions between different
integral curve Let X be a vector field on a
parts of the same molecular entity.
manifold M.An integral curve of X with initial
(2) Relating to a comparison between atoms
condition x at t = 0, is a smooth curve c : or groups within the same molecular entity.
0
[a, b] → M such that c(0) = x and
0
intramolecular isotope effect A kinetic iso-

c (t) = X(c(t)), for all t ∈ [a, b].
tope effect observed when a single substrate,
See also dynamical system. in which the isotopic atoms occupy equivalent
reactive positions, reacts to produce a nonstatis-
intension In this context only, the use of an tical distribution of isotopomeric products. In
implicit representation of a datum in a database such a case the isotope effect will favor the path-
or model, which is not fully instantiated within way with lower force constants for displacement
the database and whose value must be computed of the isotopic nuclei in the transition state.
according to the representation.
Comment: The restriction is meant to avoid intrinsic rate constant See rate constant.
philosophical wrangling and confine the dis-
course to that of databases. A common inten- intrinsic viscosity (of a polymer) The lim-
sional representation is a declarative rule, but the iting value of the reduced viscosity, η /c, or the
i
output of a statistical algorithm, federated to a inherent viscosity, η inh , at infiinite dilution of the
database and performed on data derived from it polymer, i.e.,
or other computations, is another example. See
[η] = lim(η /c) = lim η inh
i
also extension. c→0 c→0
interior (of a set) For a subset A of a topo- Notes: (1) This term is also known in the lit-
logical space, the set of points p ∈ A such that erature as the Staudinger index.
3 −1
(2) The unit must be specified; cm g is rec-
A also contains a neighborhood of p.
ommended.
intermediate A molecular entity with a life- (3) This quantity is neither a viscosity nor a
time appreciably longer than a molecular vibra- pure number. The term is to be looked on as a
tion (corresponding to a local potential energy traditional name. Any replacement by consistent
minimum of depth greater than RT ) that is terminology would produce unnecessary confu-
formed (directly or indirectly) from the reactants sion in the polymer literature. Synonymous with
and reacts further to give (either directly or indi- limiting viscosity number.
rectly) the products of a chemical reaction; also
the corresponding chemical species. inverse function theorem Let V, W be
Banach spaces, U ⊂ V open and f : U ⊂
internal energy, U U U Quantity of change V → W differentiable. If Df (x ) is a lin-
0
which is equal to the sum of heat, q, brought ear isomorphism Df (x ) : V → W, then f
0
to the system and work, w, performed on it, is a local diffeomorphism, i.e., a diffeomorphism
U = q + w. Also called thermodynamic from a neighborhood of x onto a neighborhood
0
energy. of f(x ).
0

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC