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A primary source of electromotive force is electromagnetic induction, described by Faraday's law. It states that a time-varying magnetic flux [latex]\Phi_B[/latex] through a circuit loop induces an EMF ([latex]\mathcal{E}[/latex]). The magnitude of the EMF is proportional to the rate of change of the flux, given by the equation [latex]\mathcal{E} = - \frac{d\Phi_B}{dt}[/latex]. This principle is the foundation for electric generators, transformers, and inductors.

A reformulation of classical mechanics that uses generalized coordinates and their conjugate momenta. It is based on the Hamiltonian function, [latex]H(q, p, t)[/latex], representing the system's total energy. The dynamics are described by Hamilton's equations: [latex]\dot{q}_i = \frac{\partial H}{\partial p_i}[/latex] and [latex]\dot{p}_i = -\frac{\partial H}{\partial q_i}[/latex]. This framework is central to quantum mechanics and statistical mechanics.

This law states that the mass of a substance altered at an electrode during electrolysis is directly proportional to the quantity of electricity transferred. The relationship is expressed by the formula [latex]m = \frac{Q}{F} \frac{M}{z}[/latex], where m is mass, Q is total electric charge, F is the Faraday constant, M is molar mass, and z is the valence number of ions of the substance.

A fundamental principle stating that the total energy of an isolated system remains constant over time. Energy can neither be created nor destroyed, only transformed from one form to another, such as from potential to kinetic energy. In classical mechanics, for systems with only conservative forces, the total mechanical energy [latex]E = T + V[/latex] is conserved.

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials, like a rubber band, strain when stretched and quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both.

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: [latex]\neg(P \land Q) \iff (\neg P) \lor (\neg Q)[/latex]. The second states that the negation of a disjunction is the conjunction of the negations: [latex]\neg(P \lor Q) \iff (\neg P) \land (\neg Q)[/latex].

Passivation is the process by which a material becomes 'passive,' meaning it is less affected by environmental factors such as corrosion. It involves the spontaneous formation of a very thin, non-reactive surface film that acts as a barrier, shielding the bulk material from further attack. This film is typically an oxide or nitride layer, a few nanometers thick.

A fundamental solution of a linear partial differential operator [latex]L[/latex] is a solution to the equation [latex]Lu = delta(x)[/latex], where [latex]delta(x)[/latex] 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 [latex]Lu = f(x)[/latex] can be found by convolution: [latex]u(x) = (G * f)(x)[/latex], where [latex]G[/latex] is the fundamental solution.

This law states that the electromotive force (EMF, [latex]\mathcal{E}[/latex]) induced in any closed circuit is equal to the negative of the time rate of change of the magnetic flux ([latex]\Phi_B[/latex]) through the circuit. The mathematical expression is [latex]\mathcal{E} = -\frac{d\Phi_B}{dt}[/latex]. This principle is the basis for electric generators, transformers, and inductors, describing the macroscopic effect of induction.

Lenz's law provides the direction of the induced electromotive force (EMF) and current resulting from electromagnetic induction. It states that the induced current will flow in a direction that creates a magnetic field opposing the change in magnetic flux that produced it. This principle is a consequence of the conservation of energy and is represented by the negative sign in Faraday's law.

An improvement on the Voltaic pile, the Daniell cell consists of a copper electrode in a copper(II) sulfate solution and a zinc electrode in a zinc sulfate solution, separated by a porous barrier. This two-fluid design prevents hydrogen gas buildup (polarization) on the copper electrode, resulting in a much more stable and reliable voltage source for a longer duration.

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 [latex]K[/latex] over the entire surface [latex]M[/latex] is equal to [latex]2\pi[/latex] times the Euler characteristic [latex]\chi(M)[/latex] of the surface. The formula is [latex]\int_M K \, dA = 2\pi \chi(M)[/latex].

For a Newtonian fluid, viscosity is a function of temperature and pressure but not shear rate. In liquids, viscosity decreases significantly as temperature increases because higher thermal energy allows molecules to overcome cohesive intermolecular forces more easily. Conversely, in gases, viscosity increases with temperature as more frequent molecular collisions at higher speeds lead to greater momentum transfer.

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.

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 [latex]\mathbb{R}^n[/latex]. These local coordinate systems, called charts, are related by smooth transition functions, forming an atlas that defines the manifold's differentiable structure.