Theory: Numerical Calculus for Real Systems

Note

This documentation teaches numerical differentiation from a practical perspective: approximating unknown dynamical systems from discrete, noisy observations.

The Central Thesis

We cannot compute exact derivatives from data. But by approximating the underlying system with a differentiable surrogate, we transform an intractable problem into a tractable one—trading exactness for practicality.

Chapters

Part I: Foundations

• Introduction: The Approximation Paradigm
• Chapter 1: Numerical Differentiation
• Chapter 2: Noise and Smoothing
• Chapter 3: Interpolation Methods

Part II: Extensions

• Chapter 4: Multivariate Derivatives
• Chapter 5: Approximation Theory
• Chapter 6: Differential Equations from Data
• Chapter 7: Stochastic Calculus

Part III: Practice

• Chapter 8: Applications Under Error
• Bibliography
• Theory: Numerical Calculus for Real Systems

Quick Reference

PyDelt Methods

Concept	Use Case	PyDelt Method
First derivative	Velocity, rate of change	.differentiate(order=1)
Second derivative	Acceleration, curvature	.differentiate(order=2)
Gradient (∇f)	Optimization, sensitivity	MultivariateDerivatives.gradient()
Jacobian	Vector field analysis	MultivariateDerivatives.jacobian()
Hessian	Curvature, stability	MultivariateDerivatives.hessian()
Laplacian	Diffusion, PDEs	MultivariateDerivatives.laplacian()

Who This Is For

• Data scientists who know basic calculus but need numerical methods

• Engineers working with sensor data and dynamical systems

• Researchers in physics, biology, or finance dealing with noisy observations

• ML practitioners who want to understand gradients beyond autodiff

Prerequisites: Undergraduate calculus, basic linear algebra, Python/NumPy.

Start your journey: Introduction: The Approximation Paradigm