Given a smooth point on any manifold and a smooth path in
the manifold such that , one can construct a differential
operator on the functions defined near by the rule

These differential operators at form a finite-dimensional vector space, and the dimension of this vector space is equal to the dimension of the manifold. This vector space is called the

For the Stiefel manifold, and, generally, all smooth constraint
manifolds, the tangent space at any point is easy to characterize.
The constraint equation can be thought of as
independent functions on
which must all have
constant value. If is a tangent vector of
at
, it must then be that

which is obtained by taking (where ). In a constraint manifold, the tangents at can equivalently be identified with those infinitesimal displacements of which preserve the constraint equations to first order.

For to be a tangent, its components must satisfy independent equations. These equations are all linear in the components of , and thus the set of all tangents is an -dimensional subspace of the vector space .

A differential operator may be associated with an matrix,
, given by

is the directional derivative operator in the direction. One may observe that for to be a tangent vector, the tangency condition is equivalent to .

While tangents are matrices and, therefore,
have associated differential operators,
*differentials* are something else. Given a function and
a point , one can consider the equation for
( is not necessarily tangent). This expression is linear in
and takes on some real value. It is thus possible to represent it as a
linear function on the vector space
,

for some appropriate matrix whose values depend on the first-order behavior of near . We identify this matrix with , the differential of at . This is the same differential which is computed by the

`dF`

functions in the sample problems.
For any constraint manifold the
differential of a smooth function can be computed without having
to know anything about the manifold itself. One can simply use the
differentials as computed in the ambient space (the unconstrained
derivatives in our case). If one then restricts
one's differential operators to be only those in tangent directions,
then one can still use the unconstrained in
to compute for
. This is why it
requires no geometric knowledge to produce the `dF`

functions.