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The first electric battery, it produces a direct current by stacking pairs of dissimilar metal discs (e.g., zinc and copper) separated by brine-soaked cloth. Each pair forms a galvanic cell, and stacking them in series increases the overall voltage. This arrangement demonstrated the first continuous conversion of chemical energy into electrical energy, paving the way for modern electrochemistry.

The Voltaic pile established the principle of adding voltages by connecting electrochemical cells in series. Each zinc-copper pair, or cell, generates a characteristic electromotive force (EMF) of approximately 0.76 volts. By physically stacking these cells, Volta demonstrated that the total voltage across the pile is the sum of the individual EMFs of each cell, as described by [latex]V_{total} = n \times V_{cell}[/latex].

A primary flaw of the Voltaic pile is electrode polarization. During operation, hydrogen gas produced at the copper cathode forms an insulating layer of bubbles on its surface. This layer increases the battery's internal resistance and reduces the surface area available for reaction. As a result, the voltage and current output of the pile drop significantly shortly after it begins to be used.

The Navier–Stokes equations are a set of non-linear partial differential equations describing the motion of viscous fluid substances. They are a statement of Newton's second law, balancing momentum changes with pressure gradients, viscous forces, and external forces. For an incompressible fluid, the equation is [latex]\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}[/latex].

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

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.

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.

Electromotive force ([latex]\mathcal{E}[/latex]) is the work done per unit of electric charge by a non-electrical source, such as a battery or generator. Despite its name, it is not a mechanical force but an electric potential, measured in volts. It represents the energy converted from another form (chemical, mechanical) into electrical energy as a charge traverses the source.

The electric current in a Voltaic pile is produced by a redox reaction. At the zinc anode, zinc metal is oxidized, releasing two electrons per atom ([latex]Zn \rightarrow Zn^{2+} + 2e^{-}[/latex]). These electrons travel through the external circuit to the copper cathode. There, hydrogen ions from the aqueous electrolyte are reduced, forming hydrogen gas ([latex]2H^{+} + 2e^{-} \rightarrow H_2[/latex]).

The electrolysis of water is the decomposition of water (H₂O) into its constituent elements, oxygen (O₂) and hydrogen (H₂), by passing an electric current through it. At the cathode, two water molecules are reduced to form hydrogen gas and hydroxide ions. At the anode, two water molecules are oxidized to form oxygen gas, protons, and electrons.

The continuum assumption treats fluids as continuous matter rather than as discrete molecules. This simplification is valid when the length scale of the problem is much larger than the intermolecular distance, allowing properties like density and velocity to be defined at infinitesimally small points. This enables the use of differential equations to describe the macroscopic behavior of fluid flow.

The Cauchy stress tensor, denoted [latex]\boldsymbol{\sigma}[/latex], is a second-order tensor that completely defines the state of stress at a point inside a material. It relates the traction vector (force per unit area) [latex]\mathbf{T}[/latex] on any surface passing through that point to the surface's normal vector [latex]\mathbf{n}[/latex] via the linear relationship [latex]\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}[/latex].

Cathodic protection (CP) is a technique used to control the corrosion of a metal surface by making it the cathode of an electrochemical cell. This is achieved by supplying an external electrical current that suppresses the natural corrosion currents. The protected metal's potential is polarized to a more negative value, shifting it into a region of immunity or passivity.

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.