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The covariant derivative of a section σ of a kinds of symmetry operations, screw axes and
bundle (B, M, π; F) with respect to a connec- glide planes. (cf. G.H. Stout and L.H. Jensen,
tion N is defined by X-Ray Structure Determination: A Practical
Guide, 2nd ed., John Wiley & Sons, New York,
∇ σ = T σ(X) − N(X) 1989).
X
i
µ
i
= X (∂ σ − N (x, σ))∂ i
µ
µ
curl The curl of a vector field F = (F ,
1
3
µ
where X = X ∂ is a vector field over M, T F ,F ) on R is
µ
2
3
is the tangent map and N(X) is the horizontal
curl F =∇ × F
lift of X with respect to the connection N. It is
a vertical vector field defined over the section σ, ∂F 3 ∂F 2 ∂F 1 ∂F 3
= − i + − j
i.e., a section of the bundle T B → M which ∂y ∂z ∂z ∂x

projects over the section σ : M → B. ∂F 2 ∂F 1
+ − k
∂x ∂y
covariant tensor See contravariant tensor.
curve (in a topological space X) A continu-
covering space of M M M A space C and a pro- ous map γ : R → X. Sometimes the domain
jection π : C → M which is a local diffeomor- is restricted to an interval I ⊂ R; if the origin
0 ∈ R is a point of I, then the curve is said to
phism. A covering space can be regarded as a
be based at x = γ(0).If X is a differentiable
bundle with a discrete standard fiber. Covering
manifold, the curve γ is usually required to be
spaces are characterized by the property of the
differentiable.
lifting of curves: if γ is a curve in M based at the
Not to be confused with the trajectory or path
point x ∈ M and b ∈ B is a point projecting on
which is the image of the curve, i.e., the subset
x = π(b), then there exists a unique curve ˆγ in B
.(γ ) ⊂ X of the space X.
based at b which projects over γ (t) = π( ˆγ (t)).
One can suitably define the composition of
Furthermore, the lift of two homotopic curves
two curves γ , λ :[0, 1] → X provided that
produces two curves of B which are still homo-
γ(1) = λ(0). Notice that even if the two curves
topic.
γ and λ are differentiable, the composition λ ∗ γ
can be nondifferentiable.
critical configuration See Hamilton prin-
ciple.
cycle A path through the graph G(V, E)

consisting of nodes V ={v ,v ,v ,...,v }
n
2
1
3
critical point A critical point of a vector and edges E ={(v ,v ), (v ,v ),...,(v ,v )},

n
1
3
2
1
2
field X is a point x such that X(x ) = 0. V ⊆ V, E ⊆ E. The length of the cycle is n.

0
0
Comment: It is the property of “pathness,”
crystallographic group Crystals are that there is a sequence of edges connecting the
formed with symmetries. In three-dimensional nodes in the cycle, which distinguishes a cycle
space, these symmetries are represented by from a disconnected subgraph. This is the only
the crystallographic groups. These include part of the definition which might otherwise not
rotation, reflection (point group), and translation be intuitive.
(lattice symmetry) as basic elements, and their
composition (e.g., screw axes and glide planes) cytoskeleton A dynamic polymer network
lead to a finite group. More specifically, 32 underneath the cell membrane. It is mostly
point groups combined with 14 Bravais lattices responsible for the mechanics, i.e., shape and
(7 primitive and 7 nonprimitive) result in 230 motility, of cells. The main types of filaments
unique space groups which describe the only in the network are actin filaments, microtubules,
ways in which identical objects may be arranged and intermediate filaments. There are many
in an infinite 3D lattice. A simpler, practical other different proteins involved in the network.
approach leading to the same result studies They regulate the state and dynamics of the
translations in a lattice, introducing two new network.
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC