partial differentiation


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Partial differentiation


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.


Very generally, let (X i) i(X_i)_i be a family of differentiable spaces (in some sense), let YY be another such space, and let ff be a differentiable map to YY from a subspace UU of the cartesian product iX i\prod_i X_i. Let dd be a relevant differential or derivative operator, and let x ix_i be the composite

U iX iX i U \hookrightarrow \prod_i X_i \twoheadrightarrow X_i

of the inclusion map of UU and the iith product projection (the iith coordinate). Then under good conditions, we have

df= i ifdx i d{f} = \sum_i \partial_i{f} \,d{x_i}

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

More precisely, we choose a category of differentiable spaces and differentiable maps between them, on which there is an endofunctor that takes each space UU to a notion of tangent bundle TUT{U}, which is assumed to be a vector bundle over UU, and takes a map f:UYf\colon U \to Y to df:TUTYd{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 dx i:TUTX id{x_i}\colon T{U} \to T{X_i}, if p:T x i(p)X iT f(p)Y\partial_i{f}_p\colon T_{x_i(p)}{X_i} \to T_{f(p)}{Y} is a linear operator between stalks (for pp a point in UU), and the sum takes place in the vector space T f(p)YT_{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) iI(X_i)_{i \in I} be a family of smooth spaces and YY another smooth space. For simplicity, let f: iX iYf \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= iX iU = \prod_i X_i in the above). For i 0Ii_0 \in I we can use the cartesian closed structure to define a morphism

C ( iX i,Y)C ( ii 0X i,C (X i 0,Y)). 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: iX iYf \colon \prod_i X_i \to Y we get a parametrised family of morphisms X i 0YX_{i_0} \to Y which we could write (using parameters) as f(x i 0^)(x i 0)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 0YX_{i_0} \to Y, thus yielding df i 0(x i 0^)(x i 0,v)d f_{i_0}(x_{\widehat{i_0}})(x_{i_0},v) as a morphism ii 0X iC (TX i 0,TY)\prod_{i \ne i_0} X_i \to C^\infty(T X_{i_0}, T Y). In full, df i 0d f_{i_0} is the image of ff under the chain of morphisms:

C ( iX i,Y)C ( ii 0X i,C (X i 0,Y))C ( ii 0X i,)C ( ii 0X i,C (TX i 0,TY)). 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 ff along X i 0X_{i_0}.


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

(fu) v,w,=(df) v,w,du=(df) v,w,(du) v,w,, \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 ‘dd’. (The reason for the latter equality is that ix j\partial_i{x_j} is the Kronecker delta δ i,j\delta_{i,j}.)


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)

Last revised on November 4, 2017 at 19:54:35. See the history of this page for a list of all contributions to it.