Differentiable Manifolds (geom)

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.

A fundamental solution of a linear partial differential operator \(L\) is a solution to the equation \(Lu = delta(x)\), where \(delta(x)\) is the Dirac delta function. It represents the response of the system to a point source or impulse. Once known, the solution to the inhomogeneous equation \(Lu = f(x)\) can be found by convolution: \(u(x) = (G * f)(x)\), where \(G\) is the fundamental solution.

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: \(\neg(P \lor Q) \iff (\neg P) \land (\neg Q)\).

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 theorem states that for any continuous function \(f\) mapping a compact convex set to itself, there is a point \(x_0\) such that \(f(x_0) = x_0\). This point is called a fixed point. Informally, if you take a map of a country, crumple it up, and place it inside the country's borders, there will always be at least one point on the map directly above its corresponding real-world location.

The Courant–Friedrichs–Lewy (CFL) condition is a necessary stability criterion for numerical solutions of hyperbolic partial differential equations using explicit time-integration schemes. It dictates that the time step size must be small enough that information does not travel further than one spatial grid cell per time step. For a 1D case, \(C = u \frac{\Delta t}{\Delta x} \le C_{max}\), ensuring numerical stability.

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.

A topological space is an ordered pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a collection of subsets of \(X\), called open sets, satisfying three axioms: 1) The empty set \(\emptyset\) and \(X\) itself are in \(\tau\). 2) The union of any number of sets in \(\tau\) is also in \(\tau\). 3) The intersection of any finite number of sets in \(\tau\) is also in \(\tau\).

The Finite Element Method (FEM) is a powerful numerical technique for solving complex engineering and physics problems described by partial differential equations. It works by discretizing a continuous domain into a set of smaller, simpler subdomains called 'finite elements'. This allows for the approximate numerical solution of problems in structural analysis, heat transfer, fluid flow, and electromagnetism.