nLab partial differentiation

Idea

When a multifunction is differentiated with respect to any one of its arguments alone, holding the others fixed, then we are engaged in partial differentiation.

Definition

Very generally, let $(X_i)_i$ be a family of differentiable spaces (in some sense), let $Y$ be another such space, and let $f$ be a differentiable map to $Y$ from a subspace $U$ of the cartesian product $\prod_i X_i$. Let $d$ be a relevant differential or derivative operator, and let $x_i$ be the composite

$$U \hookrightarrow \prod_i X_i \twoheadrightarrow X_i$$

of the inclusion map of $U$ and the $i$th product projection (the $i$th coordinate). Then under good conditions, we have

$$d{f} = \sum_i \partial_i{f} \,d{x_i}$$

for a unique family $(\partial_i{f})_i$ of linear operators, the partial derivatives of $f$ with respect to this decomposition of $U$. The term $\partial_i{f} \,d{x_i}$, which may be denoted $d_i{f}$, is similarly a partial differential of $f$.

More precisely, we choose a category of differentiable spaces and differentiable maps between them, on which there is an endofunctor that takes each space $U$ to a notion of tangent bundle $T{U}$, which is assumed to be a vector bundle over $U$, and takes a map $f\colon U \to Y$ to $d{f}\colon T{U} \to T{Y}$. (Note that this isn’t the case for generalised smooth spaces, but we could take microlinear spaces, as well as more familiar examples such as differentiable manifolds.) Then $d{x_i}\colon T{U} \to T{X_i}$, $\partial_i{f}_p\colon T_{x_i(p)}{X_i} \to T_{f(p)}{Y}$ is a linear operator between stalks (for $p$ a point in $U$), and the sum takes place in the vector space $T_{f(p)}{Y}$.

We can extend this if we work in a cartesian closed category of generalised smooth spaces. As in the above, let $(X_i)_{i \in I}$ be a family of smooth spaces and $Y$ another smooth space. For simplicity, let $f \colon \prod_i X_i \to Y$ be a smooth map (aka morphism in the category) defined on the whole product (so we take $U = \prod_i X_i$ in the above). For $i_0 \in I$ we can use the cartesian closed structure to define a morphism

$$C^\infty(\prod_i X_i, Y) \xrightarrow{\cong} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(X_{i_0},Y)).$$

Thus given a morphism $f \colon \prod_i X_i \to Y$ we get a parametrised family of morphisms $X_{i_0} \to Y$ which we could write (using parameters) as $f(x_{\widehat{i_0}})(x_{i_0})$. As taking the derivative is a smooth functor, we can partially differentiate the morphisms by applying differentiation to the morphisms $X_{i_0} \to Y$, thus yielding $d f_{i_0}(x_{\widehat{i_0}})(x_{i_0},v)$ as a morphism $\prod_{i \ne i_0} X_i \to C^\infty(T X_{i_0}, T Y)$. In full, $d f_{i_0}$ is the image of $f$ under the chain of morphisms:

$$C^\infty(\prod_i X_i, Y) \xrightarrow{\cong} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(X_{i_0},Y)) \xrightarrow{C^\infty(\prod_{i \ne i_0} X_i, -)} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(T X_{i_0}, T Y)).$$

This is the partial derivative of $f$ along $X_{i_0}$.

Notation

When the coordinates $x_i$ are given individual names $u, v, w, \ldots$, one usually writes $\partial{f}/\partial{u}$ for $\partial_i{f}$ (where $u$ replaces $x_i$); but $(\partial{f}/\partial{u})_{v,w,\ldots}$ is less ambiguous. Similarly, one can write $(d{f})_{v,w,\ldots}$ for the partial differential $(\partial{f}/\partial{u})_{v,w,\ldots} \,d{u}$, which is $d_i{f}$ when $u$ replaces $x_i$. (If $d{f}$ is thought of as an infinitesimal change in $f$, then $(d{f})_{v,w,\ldots}$ is an infinitesimal change subject to the condition that $v,w,\ldots$ are fixed.) Then

$$\left(\frac{\partial{f}}{\partial{u}}\right)_{v,w,\ldots} = \frac{(d{f})_{v,w,\ldots}}{d{u}} = \frac{(d{f})_{v,w,\ldots}}{(d{u})_{v,w,\ldots}} ,$$

which explains the notation and why ‘$\partial$’ looks like ‘$d$’. (The reason for the latter equality is that $\partial_i{x_j}$ is the Kronecker delta $\delta_{i,j}$.)

Related concepts

• partial differential equation

• jet bundle

• differentiable function, smooth function

• function with rapidly decreasing partial derivatives

References

The Kock-Lawvere axiom for the axiomatization of differentiation in synthetic differential geometry was introduced in

• Anders Kock, A simple axiomatics for differentiation, Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 (JSTOR)