Differentiation and Integration - Executable Engineering: Interactive Numerical Methods in Python

In continuum mechanics, dynamics, electromagnetism, and thermal transport, physical laws are formulated through differential and integral equations. Differentiation reveals local, instantaneous rates of change from positions or states. Integration accumulates rates, flux densities, and source distributions to yield net displacements, total charge, or mechanical work. In physical telemetry, both operations are indispensable.

Known equations¶

If we knew the type of equation (or had it symbolically), we could find the derivative / integral analytically.

This can be extended to function comoposition via the chain rule which is the basis of backwards differentiation in computational graphs.

Of all the families of functions, polynomials are trivial to differentiate / integrate. Fortunatley, we can always approximate the local function behaviour via a taylor expansion which will be a key tool in deriving approaches and approximating the error.

\begin{aligned} f(x + h) &= f(x) + h f'(x) + \frac{h^2}{2!} f''(x) + \frac{h^3}{3!} f'''(x) + O(h^4) \\ f(x - h) &= f(x) - h f'(x) + \frac{h^2}{2!} f''(x) - \frac{h^3}{3!} f'''(x) + O(h^4) \end{aligned}

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Algorithmic Roadmap for This Chapter¶

To equip engineers with reliable tools for both operations, this chapter is divided into:

Finite Difference Methods:
- Forward, backward, and central difference stencils from Taylor expansions.
- Second-order derivatives and Laplacian operators in higher dimensions $\nabla^2 u$ .
- Uneven grid spacing and boundary treatment.
- Truncation error versus machine precision roundoff amplification.
Numerical Integration and Quadrature:
- Newton-Cotes composite formulae (Midpoint, Trapezoid, Simpson’s $1/3$ , and $3/8$ rules).
- Richardson extrapolation and Romberg integration tableaux.
- Gaussian quadrature using Legendre polynomial roots for optimal algebraic accuracy.