nLab
partial differentiation

Partial differentiation

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          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(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}.

          Notation

          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}.)

          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)

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