Gradient Operators

📐 Definition

For $f : \mathbb{R}^d \to \mathbb{R}$ with sufficient differentiability,

\nabla f(x) = \left(\frac{\partial f}{\partial x_1}(x), \dots, \frac{\partial f}{\partial x_d}(x)\right)

Domain and Codomain

The gradient maps differentiable scalar fields on $\mathbb{R}^d$ (or a manifold) to vector fields in $\mathbb{R}^d$ (or the tangent bundle).

⚙️ Key Properties

Linearity

\nabla(af + bg) = a\,\nabla f + b\,\nabla g

Geometry of Level Sets

Where $f$ is smooth and $\nabla f \ne 0$ , the gradient is orthogonal to the level set $\{x : f(x)=c\}$ .

Weak/Distributional Gradient

In PDE settings, $\nabla$ extends via weak derivatives, allowing gradients of functions with limited regularity.

🎯 Special Cases and Limits

In one dimension, $\nabla f$ reduces to the ordinary derivative $f'$ .
On periodic domains, gradients can be represented spectrally by multiplication with $ik$ in Fourier space.

The Laplacian is the divergence of the gradient. Finite differences discretize $\nabla$ on grids, while the fractional Laplacian extends differentiation to nonlocal orders.

Usage in Oakfield

Oakfield’s built-in gradient/derivative behavior is provided by operator kernels:

Finite differences: spatial_derivative computes gradient-like quantities using stencils (real/complex component-wise paths).
Convolutional approximations: minimal_convolution and sieve can be used as smoothing prefilters before derivative estimation.
Spectral operators: while not a gradient operator per se, FFT-based pipelines allow building derivative-like multipliers in frequency space when needed.

Historical Foundations

📜 Vector Calculus

The gradient is a central object of vector calculus, connecting scalar potentials to directional rates of change and underpinning classical PDEs and continuum physics.

🌍 Modern Perspective

In numerical analysis, the gradient serves as a primitive for building divergence, Laplacian, and variational operators across discretizations.

📚 References

Evans, Partial Differential Equations
LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations