Tangent Map

If $f: M\to N$, then the tangent map $Tf$ associated to $f$ is a Vector Bundle Homeomorphism $Tf: TM \to TN$ (i.e., a Map between the Tangent Bundles of $M$ and $N$ respectively). The tangent map corresponds to Differentiation by the formula

$$Tf(v) = (f\circ\phi)'(0),$$ (1)

where $\phi'(0) = v$ (i.e., $\phi$ is a curve passing through the base point to $v$ in $TM$ at time 0 with velocity $v$). In this case, if $f: M\to N$ and $g: N\to O$, then the Chain Rule is expressed as

$$T(f\circ g) = Tf\circ Tg.$$ (2)

In other words, with this way of formalizing differentiation, the Chain Rule can be remembered by saying that ``the process of taking the tangent map of a map is functorial.'' To a topologist, the form

$$(f\circ g)'(a) = f'(g(a))\circ g'(a),$$ (3)

for all $a$, is more intuitive than the usual form of the Chain Rule.

See also Diffeomorphism

References

Gray, A. ``Tangent Maps.'' §9.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 168-171, 1993.