A Lagrangian coordinate labels a material particle and remains attached to it as the continuum moves.

An Eulerian coordinate labels a fixed position in space at which continuum fields are observed.

Substitution of $n(t)=e^{\lambda t}$ into $n^{\prime }(t)=rn(t-\tau )$ gives the transcendental characteristic equation

Given the solution on one interval of length $\tau$, the delayed term is known on the next interval, where the delay equation becomes an ordinary differential equation. Repeating this process constructs the solution interval by interval.

For a localized steady current density,It has dimensions $\mathrm{A}{\mathrm{m}}^2$.

The transformation $A\mapsto A+\nabla \chi$ leaves $B=\nabla \times A$ unchanged because the curl of a gradient vanishes.

The Coulomb gauge imposeson the magnetic vector potential. A gauge transformation $A\mapsto A+\nabla \chi$ reaches this gauge when $\chi$ solves

For thin closed wire loops $C_1,C_2$, the mutual inductance is

The magnetic flux through an oriented surface $S$ is ${\Phi }_B={\int }_{S}B\cdot dS$.

On a compact positively invariant set where $\dot{V}\le 0$, every trajectory approaches the largest invariant subset of $\{\dot{V}=0\}$.

If a Lyapunov function is nonincreasing on a sublevel set, trajectories cannot cross its boundary outward.

A positive-definite quadratic form defines ellipsoidal sublevel sets and often turns a polynomial vector field into an exact factored orbital derivative.

An equilibrium is asymptotically stable when it is Lyapunov stable and every trajectory starting sufficiently nearby converges to it.

A positive-definite Lyapunov function with strictly negative orbital derivative away from the equilibrium proves asymptotic stability.

A positive-definite Lyapunov function with nonpositive orbital derivative proves Lyapunov stability of the equilibrium. Indeed, fix a sufficiently small ball $B_{\epsilon }$ in the function's domain and putContinuity and $V(0)=0$ give a ball $B_{\delta }$ on which $V<\alpha$. Since $V$ cannot increase along a trajectory, one starting in $B_{\delta }$ cannot cross the sphere $\Vert x\Vert =\epsilon$, on which $V\ge \alpha$. It therefore remains in $B_{\epsilon }$ for all forward time.

An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.

Expansion reduces bolometric flux by one factor of $1+z$ from photon energy and one from arrival rate, in addition to inverse-area dilution.

For a boost of speed $V$ along $x$,

The nonrelativistic limit takes characteristic speeds much smaller than $c$. Lorentz transformations then reduce to Galilean transformations at leading order.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.