Graphics Reference
In-Depth Information
Vect (
M
)
= vector space of vector fields on the manifold
M
= signal curvature function of curve in
R
2
k
S
(s)
k(s)
= curvature function
t(s)
= torsion function
<s>
= simplicial complex generated by simplex s
[s]
= oriented simplex s
C
q
(K)
= group of q-chains
B
q
(K)
= group of q-boundaries
Z
q
(K)
= group of q-cycles
H
q
(K)
= q-th homology group
ª
= homeomorphic, isomorphic
= homotopic, homologous
A
= homotopic relative to
A
Partial derivative notation for a function f:
∂
∂
f
x
∂
∂
f
y
,
or f
,
f
for D f D f
,
,
respectively, etc,
xy
1
2
2
2
∂
∂
f
∂
∂∂
f
xy
or f
,
,
f
for D
f D
,
f respectively etc
,
,
.
xx
yy
11
,
1 2
,
2
x
T(
A
,
B
,...) = (
A
¢,
B
¢, . . .) : This means that T(
A
) =
A
¢, T(
B
) =
B
¢,...
Commutative diagram:
In general, if one has a directed graph where the nodes are
sets and the arrows correspond to maps between these sets, then this is said to con-
stitute a commutative diagram if, whenever two directed paths start and end at the
same points, the corresponding composition of maps is equal. Commutative diagrams
are nice to have and the terminology is useful in many areas of mathematics. As an
example, consider the diagram
f
A
ææ
B
g
Ø
Ø
F
C
ææ
D
G
If G(g(a)) = F(f(a)) for all a ΠA, then the diagram is said to be
commutative
.
