Shape Derivatives

Abstract: Shape Theory, together with Shape-and-Scale Theory, comprise Relational Theory. This consists of $N$-point models on a manifold $M$, for which some geometrical automorphism group $G$ is regarded as meaningless and is thus quotiented out from the $N$-point model's product space $\times_{I = 1}^N M$. Each such model has an associated function space of preserved quantities, solving the PDE system for zero brackets with the sums over $N$ of each of $G$'s generators. These are smooth functions of the $N$-point geometrical invariants. Each $(M, G)$ pair has moreover a `minimal nontrivially relational unit' value of $N$; we now show that relationally-invariant derivatives can be defined on these, yielding the titular notions of shape(-and-scale) derivatives. We obtain each by Taylor-expanding a functional version of the underlying geometrical invariant, and isolating a shape-independent derivative factor in the nontrivial leading-order term. We do this for translational, dilational, dilatational and projective geometries in 1-$d$, the last of which gives a shape-theoretic rederivation of the Schwarzian derivative. We next phrase and solve the ODEs for zero and constant values of each derivative. We then consider translational, dilational, rotational, rotational-and-dilational, Euclidean and equi-top-form (alias unimodular affine) cases in $\geq 2$-$d$. We finally pose the PDEs for zero and constant values of each of our $\geq 2$-$d$ derivatives, and solve a subset of these geometrically-motivated PDEs. This work is significant for Relational Motion and Background Independence in Theoretical Physics, and foundational for both Flat and Differential Geometry.