Section 7 9 October 2026

Whole-form spatial and source derivatives

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23259571

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7 Whole-form spatial and source derivatives

Temporal smoothing and spatial differentiation make different demands. The preceding theorems do not require a spatial derivative of the forcing when their temporal or resolvent assumptions are earned. A physical source derivative, however, must act on the complete coupled form and the actual input; replacing that coordinate by a classical label changes the problem.

Proposition 7.1 (Ordered whole-form derivative)

Let $A$ and $\mathcal L=LF^{-1}$ be as in Theorem 6.1. Let $\delta_i$ be either a parameter derivative on the common form domain, or a closed physical derivation $\delta_iM=D_i^QM-MD_i^P$ . For a parameter family, require strong difference-quotient convergence of $A$ in $\mathcal L(\mathcal V,\mathcal V^*)$ and of $\mathcal L$ in $\mathcal L(\mathcal U,\mathcal V^*)$ , with locally uniform bounds on these quotients and uniformly equivalent nearby form norms. For a physical derivation, require domain-preserving translations or local gauges for which the conjugated families satisfy these same conditions and realize the displayed derivatives on a common core. Suppose the actual whole-form sandwiches are bounded:

\|K^{-1/2}(\delta_iA)K^{-1/2}\|\le\alpha_i,\qquad \|K^{-1/2}\delta_i\mathcal L\|\le\beta_i .

(7.1)

For $\zeta=E+i\eta$ , $\eta>0$ , write $R_\zeta=(A-\zeta)^{-1}$ and $U_\zeta=K^{1/2}R_\zeta\mathcal L$ . Then

\begin{align}\delta_i(R_\zeta\mathcal L) &=R_\zeta\{\delta_i\mathcal L -(\delta_iA)R_\zeta\mathcal L\}, \tag{7.2}\\ \|K^{1/2}\delta_i(R_\zeta\mathcal L)\| &\le J_\zeta \{\beta_i+\alpha_i\|U_\zeta\|\},\qquad J_\zeta=\sup_{\lambda\in\sigma(A)} \frac{\lambda+c}{|\lambda-\zeta|}. \tag{7.3}\end{align}

If $\zeta$ itself varies, add $(\delta_i\zeta)R_\zeta^2\mathcal L$ to (7.2).

Proof

For a parameter family, the form resolvent difference identity is

R_{\zeta,\epsilon}-R_\zeta =-R_{\zeta,\epsilon}(A_\epsilon-A)R_\zeta.

It is a bounded identity from $\mathcal V^*$ to $\mathcal V$ . Strong differentiation on fixed vectors, uniform local form bounds and the given difference-quotient hypothesis yield $\delta_iR_\zeta=-R_\zeta(\delta_iA)R_\zeta$ . Leibniz gives (7.2). For a physical derivation, conjugate by the domain-preserving translations/gauges first and apply the same argument; differentiation on the common core is the commutator formula. Strong form convergence and closedness of the physical derivative pass it to its admitted domain.

Insert a form Riesz map without changing order:

K^{1/2}R_\zeta(\delta_iA)R_\zeta\mathcal L =\big[K^{1/2}R_\zeta K^{1/2}\big] \big[K^{-1/2}(\delta_iA)K^{-1/2}\big]U_\zeta .

The first factor has norm $J_\zeta$ by the spectral theorem. The analogous factorization of the $\delta_i\mathcal L$ term proves (7.3). Differentiating $A-\zeta$ when $\zeta$ varies gives the stated extra term with its positive sign.

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For a literal input vector $a$ , the corresponding physical derivative remains

D_i^Q(R_\zeta\mathcal L a) =[\delta_i(R_\zeta\mathcal L)]a +R_\zeta\mathcal L D_i^Pa .

(7.4)

Its second term cannot be erased by a response norm estimate. The graph and derivative domain of $a$ must be included. When $F$ is covariantly constant, $\delta_i\mathcal L=(\delta_iL)F^{-1}$ ; otherwise

\delta_i\mathcal L =(\delta_iL)F^{-1} -LF^{-1}(\delta_iF)F^{-1}

on the admitted graph, with an additional bound for that second term. Gauge curvature, moving frames, source kinetic terms and mobility commutators all belong to the whole forms in (7.1). Differentiating a singular factor $W$ is unnecessary and may be invalid even when the derivative of the whole form $W^*JW$ is bounded.

For example, in three relative dimensions the Hardy inequality gives the legitimate whole Coulomb force form

|\langle u,\partial_{R_i}(g/|r-R|)v\rangle| \le |g|\|u/|r-R|\|\|v/|r-R|\| \le4|g|\|\nabla_r u\|\|\nabla_r v\|.

With an earned $K\ge\lambda_{\rm rel}(-\Delta_r)$ , its relative coefficient is at most $4|g|/\lambda_{\rm rel}$ . The coefficient is the true relative-mass kinetic coefficient; source-dependent center maps add their actual derivatives. This is a form estimate, not an $L^2$ operator bound on the force. Strong vector continuity or the stated difference-quotient contract suffices; operator-norm continuity under translations is not assumed.

All statements above keep a fixed physical block chart. For a varying projector one must include its domain-preserving transport and the moving frame term $-iU^*\dot U$ . A nonlocal Hilbert-space band unitary is not automatically an isometry of pointwise physical currents. Likewise a moving hard-core boundary requires a proved domain transport before it enters a fixed-domain force identity.