Crate amari_calculus 

Amari Calculus

Geometric calculus - a unified framework for differential and integral calculus using geometric algebra.

Overview

This crate provides geometric calculus operations that unify:

• Vector calculus (gradient, divergence, curl)
• Differential forms
• Tensor calculus
• Covariant derivatives on manifolds

Mathematical Foundation

Geometric calculus is built on the vector derivative operator:

∇ = e^i ∂_i  (sum over basis vectors)

This operator combines:

• Dot product → divergence (∇·F)
• Wedge product → curl (∇∧F)
• Full geometric product → complete derivative (∇F = ∇·F + ∇∧F)

Key Features

• Vector Derivative Operator: The fundamental ∇ operator
• Classical Operators: Gradient, divergence, curl, Laplacian
• Manifold Calculus: Covariant derivatives, connections, geodesics
• Lie Derivatives: Derivatives along vector fields
• Integration: Integration on manifolds using amari-measure
• Fundamental Theorem: ∫V (∇F) dV = ∮∂V F dS

Examples

Gradient of a scalar field

use amari_calculus::{ScalarField, VectorDerivative, CoordinateSystem};
use amari_core::Multivector;

// Define scalar field f(x, y) = x² + y²
let f = ScalarField::<3, 0, 0>::new(|coords| {
    coords[0].powi(2) + coords[1].powi(2)
});

// Create vector derivative operator
let nabla = VectorDerivative::<3, 0, 0>::new(CoordinateSystem::Cartesian);

// Compute gradient at point (1, 2)
let grad_f = nabla.gradient(&f, &[1.0, 2.0, 0.0]);

// Gradient should be approximately (2, 4, 0)

Divergence of a vector field

use amari_calculus::{VectorField, VectorDerivative, CoordinateSystem, vector_from_slice};

// Define vector field F(x, y, z) = (x, y, z)
let f = VectorField::<3, 0, 0>::new(|coords| {
    vector_from_slice(&[coords[0], coords[1], coords[2]])
});

let nabla = VectorDerivative::<3, 0, 0>::new(CoordinateSystem::Cartesian);

// Compute divergence (should be 3)
let div_f = nabla.divergence(&f, &[1.0, 1.0, 1.0]);

Curl of a vector field

use amari_calculus::{VectorField, VectorDerivative, CoordinateSystem, vector_from_slice};

// Define vector field F(x, y, z) = (-y, x, 0) (rotation around z-axis)
let f = VectorField::<3, 0, 0>::new(|coords| {
    vector_from_slice(&[-coords[1], coords[0], 0.0])
});

let nabla = VectorDerivative::<3, 0, 0>::new(CoordinateSystem::Cartesian);

// Compute curl (should be (0, 0, 2) bivector representing rotation)
let curl_f = nabla.curl(&f, &[0.0, 0.0, 0.0]);

Re-exports

pub use fields::MultivectorField;

pub use fields::ScalarField;

pub use fields::VectorField;

pub use operators::curl;

pub use operators::divergence;

pub use operators::gradient;

pub use operators::laplacian;

pub use manifold::RiemannianManifold;

Modules

fields

Field types for geometric calculus

manifold

Manifold calculus - covariant derivatives and connections

operators

Classical differential operators (gradient, divergence, curl, Laplacian)

prelude

Prelude module for convenient imports

Structs

LieDerivative

Lie derivative operator

ManifoldIntegrator

Integrator for scalar and multivector fields on manifolds

VectorDerivative

Vector derivative operator ∇

Enums

CalculusError

Errors that can occur during calculus operations

CoordinateSystem

Coordinate systems for differential operators

Functions

vector_from_slice

Utility function to create a vector multivector from a slice of components

Type Aliases

CalculusResult

Result type for calculus operations