Differential (infinitesimal) - Differentials As Linear Maps

Differentials As Linear Maps

There is a simple way to make precise sense of differentials by regarding them as linear maps. One way to explain this point of view is to regard the variable x in an expression such as f(x) as a function on the real line, the standard coordinate or identity map, which takes a real number p to itself (x(p) = p): then f(x) denotes the composite f ∘ x of f with x, whose value at p is f(x(p)) = f(p). The differential df is then a function on the real line whose value at p (usually denoted df_p) is not a number, but a linear map from R to R. Since a linear map from R to R is given by a 1×1 matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of df_p as an infinitesimal and compare it with the standard infinitesimal dx_p, which is again just the identity map from R to R (a 1×1 matrix with entry 1). It may seem fanciful to regard the identity map as an infinitesimal, but it does at least have the property that if ε is very small, then dx_p(ε) is very small. The differential df_p has the same property, because it is just a multiple of dx_p, and this multiple is the derivative f ′(p) by definition. We therefore obtain that df_p = f ′(p) dx_p, and hence df = f ′ dx. Thus we recover the idea that f ′ is the ratio of the differentials df and dx.

This would just be a trick were it not for the fact that:

1. it captures the idea of the derivative of f at p as the best linear approximation to f at p;
2. it has many generalizations.

For instance if f is a function from Rn to R then we say f is differentiable at p ∈ Rn if there is a linear map df_p from Rn to R such that for any ε > 0, there is a neighbourhood N of p such that for x ∈ N:

We can now use the same trick as in the one dimensional case, and think of the expression f(x1, x2, …, xn) as the composite of f with the standard coordinates x1, x2, …, and xn on Rn (so that xj(p) is the j-th component of p ∈ Rn). Then the differentials (dx1)_p, (dx2)_p, (dxn)_p (at a point p) form a basis for the vector space of linear maps from Rn to R and therefore, if f is differentiable at p, we can write df_p as a linear combination of these basis elements:

The coefficients D_jf(p) are (by definition) the partial derivatives of f at p with respect to x1, x2, …, and xn. Hence, if f is differentiable on all of Rn, we can write, more concisely:

In the one-dimensional case this becomes

as before.

This idea generalizes straightforwardly to functions from Rn to Rm. Furthermore, it has the decisive advantage over other definitions of the derivative that it is invariant under changes of coordinates. This means that the same idea can be used to define the differential of smooth maps between smooth manifolds.

Aside: Note that the existence of all the partial derivatives of f(x) at x is a necessary condition for the existence of a differential at x. However it is not a sufficient condition. For counterexamples, see Gâteaux derivative.