Differential in geometric algebra: numerical method

Generalized Differential in Geometric Algebra via Oriented Volume

In geometric algebra (GA), the differential can be expressed using oriented volume and the geometric product. This approach leverages finite differences on a grid to interpolate function values at arbitrary points near a node. The method decomposes the increment into components parallel and orthogonal to the gradient, making it highly useful for numerical computations.

The technique relies on matrix representations of axes using Pauli matrices and a pseudoscalar. It enables calculations without requiring additional grids or runtime matrix multiplications.

Basic Notation

• Δ — Finite increments on the grid: steps Δx, Δy, Δz and function differences Δx f, Δy f, Δz f.
• δ — Arbitrary increments in the neighborhood of a node.
• σ1, σ2, σ3 — Axis unit vectors in the Pauli matrix basis:

```math

σ1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad σ2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad σ3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}

```

• ⋅ — Geometric (matrix) product.
• × — Cross product.
• I = σ1 ⋅ σ2 ⋅ σ3 — Pseudoscalar.

A vector X in this basis is expressed as: X = x σ3 + y σ2 + z σ1. Note the permuted axes: σ3 corresponds to x, σ2 to y, and σ1 to z.

Oriented Volume and Relative Volume

The oriented volume δV is constructed from the cell face areas and displacements:

δV = ΔSx ⋅ (δx ⋅ σ3) + ΔSy ⋅ (δy ⋅ σ2) + ΔSz ⋅ (δz ⋅ σ1)

where ΔSx = Δy Δz, ΔSy = Δx Δz, and ΔSz = Δx Δy.

The relative oriented volume (obtained by dividing by the cell volume Δv = Δx Δy Δz):

δV' = \frac{δV}{Δv} = \frac{Δx}{δx} ⋅ σ3 + \frac{Δy}{δy} ⋅ σ2 + \frac{Δz}{δz} ⋅ σ1

The finite difference vector Δ(f) (analogous to an unnormalized gradient):

Δ(f) = Δx f σ3 + Δy f σ2 + Δz f σ1

with Δx f = f(x+Δx, y, z) - f(x, y, z), and similarly for the other axes.

Scalar and Bivector Increments

The scalar increment δf∥ (parallel component):

δf∥ = \frac{Δx f}{Δx} δx + \frac{Δy f}{Δy} δy + \frac{Δz f}{Δz} δz

As Δ, δ → 0, this converges to the classical differential df.

The generalized differential δg f as the geometric product δV' ⋅ Δ(f):

δg f = δf∥ + I δf⊥

The bivector component δf⊥:

δf⊥ = (Δy f \frac{δz}{Δz} - Δz f \frac{δy}{Δy}) σ3 + (Δx f \frac{δz}{Δz} - Δz f \frac{δx}{Δx}) σ2 + (Δx f \frac{δy}{Δy} - Δy f \frac{δx}{Δx}) σ1

δf⊥ = 0 if and only if δV' is parallel to Δ(f), meaning the displacement (δx, δy, δz) is collinear with the gradient. Otherwise, the bivector encodes:

• The normal to the plane spanned by δV' and Δ(f).
• An orthogonal object within that plane.
• Together with δf∥, it forms a structure analogous to a complex number.

Advantages of the Method

• Off-Grid Interpolation: From the values Δx f, Δy f, Δz f at a node, δg f can be computed for any (δx, δy, δz) in its vicinity.
• Component Separation: δf∥ represents the change along the displacement, while δf⊥ captures deviation from the gradient.
• Gradient Trajectories: The condition δf⊥ = 0 defines directions parallel to Δ(f).
• Computational Efficiency: Relies solely on scalar operations, eliminating matrices during computation.

Calculation Example

At node (0,0,0): Δx = Δy = Δz = 0.1, Δx f = 0.2, Δy f = 0.3, Δz f = 0.4 (corresponding to derivatives 2, 3, 4).

For point (0.02, 0.03, 0.04):

• δf∥ = (0.2/0.1)0.02 + (0.3/0.1)0.03 + (0.4/0.1)*0.04 = 0.29
• δf⊥ coefficients: c3 = 0.3(0.04/0.1) - 0.4(0.03/0.1) = 0; similarly, c2 = 0, c1 = 0.

The displacement is parallel to the gradient, yielding δg f = 0.29.

For point (0.03, 0.02, 0.04):

• δf∥ = 20.03 + 30.02 + 4*0.04 = 0.28
• c3 = 0.30.4 - 0.40.2 = 0.04; c2 = 0.40.3 - 0.20.4 = 0.04; c1 = 0.20.2 - 0.30.3 = -0.05.

The bivector is non-zero, indicating the method successfully captures the deviation.

Key Takeaways

• The method expresses the differential through relative oriented volume and finite differences.
• Enables interpolation at arbitrary points without auxiliary grids.
• The bivector component quantitatively measures the non-collinearity between displacement and gradient.
• Applicable in numerical methods with adaptive meshes for estimating increments along arbitrary directions.
• Computations reduce entirely to scalar arithmetic.

— Editorial Team