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### The Difference Between a Tangent and a Differential

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 tangent space of at , and is often denoted .

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.

Next: Inner Products, Gradients, and Up: Geometric Technicalities Previous: Manifolds   Contents   Index
Susan Blackford 2000-11-20