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Energy is not continuous but comes in discrete packets called quanta. The energy \(E\) of a single quantum of electromagnetic radiation (a photon) is directly proportional to its frequency \(\nu\). This relationship is defined by the Planck-Einstein relation, \(E = h\nu\), where \(h\) is the Planck constant. This concept fundamentally challenged classical physics.

A statistical ensemble is a conceptual tool consisting of a large number of virtual copies of a system, each representing a possible microstate. By averaging properties over all systems in the ensemble, one can calculate macroscopic observables. The main types are the microcanonical (isolated system with fixed N, V, E), canonical (closed system, fixed N, V, T), and grand canonical (open system, fixed µ, V, T).

The equivalence principle is a cornerstone of general relativity. It posits that the effects of a uniform gravitational field are indistinguishable from the effects of uniform acceleration. This means that an observer in a windowless, free-falling elevator experiences weightlessness, just like an observer in deep space, far from any gravitational source, forming the conceptual basis for gravity as a geometric phenomenon.

A key prediction of general relativity is that time passes at different rates in different gravitational potentials. A clock in a stronger gravitational field (closer to a massive object) will tick slower than a clock in a weaker field. This effect, known as gravitational time dilation, has been experimentally verified and is a crucial factor in modern technology like GPS.

The Einstein Field Equations (EFE) are a set of ten coupled, non-linear partial differential equations that form the core of general relativity. They describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. The equation is concisely written as \(G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}\), relating spacetime geometry to its energy-momentum content.

General relativity predicts that a massive, rotating object should 'drag' the fabric of spacetime around with it, a phenomenon known as frame-dragging or the Lense-Thirring effect. This means a gyroscope orbiting the rotating body will precess, not because of any applied torque, but because spacetime itself is being twisted by the body's rotation.

Redox (reduction-oxidation) reactions involve a transfer of electrons between chemical species. One species undergoes oxidation (loses electrons), while another undergoes reduction (gains electrons). These two processes always occur simultaneously. The species that loses electrons is the reducing agent, and the one that gains electrons is the oxidizing agent. This fundamental concept underpins electrochemistry and many biological processes.

The absolute electrochemical potential of a single electrode cannot be measured. Instead, potentials are measured relative to a universally accepted reference. The Standard Hydrogen Electrode (SHE) is the primary reference, assigned a potential of exactly 0 volts under standard conditions (1 M \(H^+\) concentration, 1 bar \(H_2\) gas, 298 K). All other standard electrode potentials are reported relative to this zero point.

The Kutta-Joukowski theorem quantifies the lift force generated by an airfoil. It states that the lift per unit span (\(L'\)) is directly proportional to the fluid density (\(\rho\)), the free-stream velocity (\(V\)), and the circulation (\(\Gamma\)) around the body: \(L' = \rho V \Gamma\). This links the abstract concept of circulation to the physical force of lift.

The boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Introduced by Ludwig Prandtl, this concept simplifies fluid dynamics problems by dividing the flow into two regions: the thin boundary layer where viscosity dominates and the outer region where inviscid flow theory can be applied.

The von Mises yield criterion predicts that yielding of a ductile material begins when the second deviatoric stress invariant, \(J_2\), reaches a critical value. It is often expressed in terms of the von Mises stress, \(\sigma_v\), a scalar value that must be less than the material's yield strength, \(\sigma_y\). Yielding occurs when \(\sigma_v = \sigma_y\).

General relativity provided the first accurate explanation for the anomalous precession of Mercury's perihelion. Newtonian gravity could not fully account for the slow, gradual shift in the orientation of Mercury's elliptical orbit. Einstein's theory correctly predicted the missing 43 arcseconds per century, attributing it to the curvature of spacetime around the Sun, a major early triumph for the theory.

A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even light—can escape. The boundary of this region is called the event horizon. Predicted by general relativity as a solution to the field equations, a black hole is the result of the complete gravitational collapse of a massive star.

General relativity predicts that the path of light is bent by gravity. When light from a distant source passes a massive object like a galaxy or a star, its path is deflected. This phenomenon, known as gravitational lensing, can magnify, distort, or create multiple images of the background source, acting like a cosmic telescope for observing the distant universe.

Complex redox reactions can be balanced using the half-reaction method. The overall reaction is split into two separate half-reactions: one for oxidation and one for reduction. Each half-reaction is balanced for atoms and charge independently, often by adding H+, OH-, and H2O in aqueous solutions. Finally, the electron counts are equalized and the half-reactions are combined.