<s>
In	O
mathematics	O
,	O
an	O
infinitesimal	B-Algorithm
transformation	I-Algorithm
is	O
a	O
limiting	B-Algorithm
form	O
of	O
small	O
transformation	B-Algorithm
.	O
</s>
<s>
This	O
is	O
conventionally	O
represented	O
by	O
a	O
3×3	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
A	O
.	O
</s>
<s>
A	O
comprehensive	O
theory	O
of	O
infinitesimal	B-Algorithm
transformations	I-Algorithm
was	O
first	O
given	O
by	O
Sophus	O
Lie	O
.	O
</s>
<s>
The	O
properties	O
of	O
an	O
abstract	O
Lie	O
algebra	O
are	O
exactly	O
those	O
definitive	O
of	O
infinitesimal	B-Algorithm
transformations	I-Algorithm
,	O
just	O
as	O
the	O
axioms	O
of	O
group	O
theory	O
embody	O
symmetry	O
.	O
</s>
<s>
The	O
term	O
"	O
Lie	O
algebra	O
"	O
was	O
introduced	O
in	O
1934	O
by	O
Hermann	O
Weyl	O
,	O
for	O
what	O
had	O
until	O
then	O
been	O
known	O
as	O
the	O
algebra	O
of	O
infinitesimal	B-Algorithm
transformations	I-Algorithm
of	O
a	O
Lie	O
group	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
case	O
of	O
infinitesimal	O
rotations	O
,	O
the	O
Lie	O
algebra	O
structure	O
is	O
that	O
provided	O
by	O
the	O
cross	O
product	O
,	O
once	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
has	O
been	O
identified	O
with	O
a	O
3-vector	O
.	O
</s>
<s>
The	O
earliest	O
example	O
of	O
an	O
infinitesimal	B-Algorithm
transformation	I-Algorithm
that	O
may	O
have	O
been	O
recognised	O
as	O
such	O
was	O
in	O
Euler	O
's	O
theorem	O
on	O
homogeneous	O
functions	O
.	O
</s>
<s>
This	O
setting	O
is	O
typical	O
,	O
in	O
that	O
there	O
is	O
a	O
one-parameter	O
group	O
of	O
scalings	B-Algorithm
operating	O
;	O
and	O
the	O
information	O
is	O
coded	O
in	O
an	O
infinitesimal	B-Algorithm
transformation	I-Algorithm
that	O
is	O
a	O
first-order	O
differential	O
operator	O
.	O
</s>
<s>
is	O
an	O
operator	O
version	O
of	O
Taylor	O
's	O
theorem	O
and	O
is	O
therefore	O
only	O
valid	O
under	O
caveats	O
about	O
f	O
being	O
an	O
analytic	B-Language
function	I-Language
.	O
</s>
<s>
Concentrating	O
on	O
the	O
operator	O
part	O
,	O
it	O
shows	O
that	O
D	O
is	O
an	O
infinitesimal	B-Algorithm
transformation	I-Algorithm
,	O
generating	O
translations	O
of	O
the	O
real	O
line	O
via	O
the	O
exponential	O
.	O
</s>
