Fundamental Solution (Green’s Function)

A second-order linear hyperbolic partial differential equation that governs the propagation of various types of waves. In its simplest form, it is written as \(\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u\), where \(u(\vec{x},t)\) is the amplitude of the wave, \(c\) is the wave speed, and \(\nabla^2\) is the Laplace operator. It models phenomena like vibrating strings, sound waves, and light waves.

The Euler characteristic is a topological invariant, a number that describes a topological space's structure or shape regardless of how it is bent. For polyhedra, it is defined by the formula \(\chi = V - E + F\), where V, E, and F are the number of vertices, edges, and faces, respectively. For a sphere, \(\chi = 2\), while for a torus, \(\chi = 0\).

A fundamental second-order linear parabolic partial differential equation describing heat distribution or other diffusion processes. Its canonical form is \(\frac{partial u}{partial t} = \alpha \nabla^2 u\), where \(u(\vec{x},t)\) is temperature, \(t\) is time, and \(\alpha\) is thermal diffusivity. Solutions model how an initial temperature distribution evolves, smoothing out irregularities over time and approaching a steady state.

De Morgan's laws are a pair of transformation rules in Boolean algebra that are fundamental to digital circuit design. The first law states that the negation of a conjunction is the disjunction of the negations: \(\neg(P \land Q) \iff (\neg P) \lor (\neg Q)\). The second states that the negation of a disjunction is the conjunction of the negations: \(\neg(P \lor Q) \iff (\neg P) \land (\neg Q)\).

A differentiable manifold is a topological space that is locally similar to Euclidean space, allowing calculus to be applied. Each point has a neighborhood that is homeomorphic to an open subset of \(\mathbb{R}^n\). These local coordinate systems, called charts, are related by smooth transition functions, forming an atlas that defines the manifold's differentiable structure.

Digital electronics is based on Boolean algebra, a mathematical system of logic introduced by George Boole. It uses two values, typically 0 and 1 (or false and true), and three basic operations: AND (conjunction), OR (disjunction), and NOT (negation). These operations directly correspond to the logic gates that form the building blocks of all digital circuits.

This is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The problem asked if the seven bridges of the city of Königsberg could all be traversed in a single trip without doubling back, with the trip ending on the same landmass it began.

A fundamental theorem in topology and geometry stating that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the formula \(V - E + F = 2\). This value, 2, is the Euler characteristic of a sphere, revealing a deep topological property independent of the polyhedron's specific shape.

A second-order linear elliptic partial differential equation that describes systems in a steady-state or equilibrium condition. It is written as \(nabla^2 u = 0\) or \(Delta u = 0\), where \(nabla^2\) (or \(Delta\)) is the Laplace operator. Solutions, called harmonic functions, are the smoothest possible functions and represent potentials in fields like electrostatics, gravitation, and fluid flow.

Theorema Egregium (Latin for "Remarkable Theorem") states that the Gaussian curvature of a surface is an intrinsic property. This means it depends only on how distances are measured on the surface itself, not on how the surface is embedded in three-dimensional space. A flat sheet of paper can be rolled into a cylinder but not a sphere without stretching.

The Gauss-Bonnet theorem connects the geometry of a compact two-dimensional surface to its topology. It states that the integral of the Gaussian curvature \(K\) over the entire surface \(M\) is equal to \(2\pi\) times the Euler characteristic \(\chi(M)\) of the surface. The formula is \(\int_M K \, dA = 2\pi \chi(M)\).

Accuracy refers to the closeness of a measured value to a standard or known true value. Precision refers to the closeness of two or more measurements to each other. A measurement system can be precise but not accurate, accurate but not precise, neither, or both. These two concepts are independent of each other in measurement theory.

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds—smooth manifolds equipped with a Riemannian metric. This metric is a collection of inner products on the tangent spaces, varying smoothly from point to point. It allows for the definition of local geometric notions like angle, length of curves, surface area, and volume, leading to a generalized notion of curvature.

A homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. Two topological spaces are called homeomorphic if such a function exists. From a topological viewpoint, homeomorphic spaces are identical. This concept captures the idea that an object can be stretched, bent, or deformed into another without tearing or gluing, like a coffee mug into a donut.