Warp Gradients

📐 Definition

For a differentiable warp $w : \mathbb{R}^d \to \mathbb{R}^d$, the warp gradient (Jacobian) is

$$J_w(x) = \nabla w(x) = \left[\frac{\partial w_i}{\partial x_j}(x)\right]_{i,j=1}^{d}$$

Copied

Domain and Codomain

Defined wherever $w$ is differentiable. Input $x \in \mathbb{R}^d$; output is a $d \times d$ matrix.

---

⚙️ Key Properties

Determinant $\det J_w$ measures local volume change; singular values quantify local stretching. Composition satisfies $J_{w \circ v}(x) = J_w(v(x))\,J_v(x)$.

---

For linear warps $w(x) = Ax$, $J_w = A$ everywhere. For conformal maps in $d=2$, $J_w$ is a scaled rotation when $w$ is analytic.

---

🎯 Special Cases and Limits

• Linear warp: constant Jacobian.
• Conformal 2D warp: Jacobian is a scaled rotation away from critical points.

---

🔗 Related Functions

Analytic warp maps provide concrete $w$; phase-space warps extend the Jacobian to canonical coordinates; hyperexponential warps furnish scalar factors embedded in $w$.

---

Usage in Oakfield

Oakfield computes “warp gradients” in the concrete sense of profile gradient sampling for analytic warp updates:

• Warp safety layer (warp_safety) samples profile gradients and applies continuity/clamp policies when the raw evaluation is invalid.
• Analytic warp profiles supply gradients for digamma/trigamma/tanh/power/hyperexp/qhyperexp paths; these gradients drive the per-step warp response.
• Diagnostics: continuity counters distinguish operators that wrote with clamped/limited guards versus unconstrained writes (surfaced in field stats).

---

Historical Foundations

📜 Jacobians and Change of Variables

Jacobians quantify local linearization and volume distortion under mappings, underpinning change-of-variables formulas and differential geometry.

🌍 Modern Perspective

Warp gradients are essential for transforming PDE operators, densities, and metrics under coordinate remapping.

---

📚 References

• Evans, Partial Differential Equations
• Lee, Introduction to Smooth Manifolds