Laplace equation and Faraday's lines of force

Boundary‐value problems involve two dependent variables: a potential function and a stream function. They can be approached in two mutually independent ways. The first, introduced by Laplace, involves spatial gradients at a point. Inspired by Faraday, Maxwell introduced the other, visualizing the flow domain as a collection of flow tubes and isopotential surfaces. Boundary‐value problems intrinsically entail coupled treatment (or, equivalently, optimization) of potential and stream functions. Historically, potential theory avoided the cumbersome optimization task through ingenious techniques such as conformal mapping and Green's functions. Laplace's point‐based approach and Maxwell's global approach each provides its own unique insights into boundary‐value problems. Commonly, Laplace's equation is solved either algebraically or with approximate numerical methods. Maxwell's geometry‐based approach opens up novel possibilities of direct optimization, providing an independent logical basis for numerical models, rather than treating them as approximate solvers of the differential equation. Whereas points, gradients, and Darcy's law are central to posing problems on the basis of Laplace's approach, flow tubes, potential differences, and the mathematical form of Ohm's law are central to posing them in natural coordinates oriented along flow paths. Besides being of philosophical interest, optimization algorithms can provide advantages that complement the power of classical numerical models. In the spirit of Maxwell, who eloquently spoke for a balance between abstract mathematical symbolism and observable attributes of concrete objects, this paper is an examination of the central ideas of the two approaches, and a reflection on how Maxwell's integral visualization may be practically put to use in a world of digital computers.