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Fermat-type optical paths in cylindrical coordinates extremize refractive index times Euclidean arclength.

Among functions with fixed boundary values, the solution of $\nabla \cdot (\kappa \nabla \varphi )=0$ with $\kappa >0$ uniquely minimizes the weighted energy

A point is stationary for a differentiable function or functional when its first derivative or first variation vanishes in every admissible direction.

A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.

The direct method takes a minimizing sequence, extracts a subsequence using compactness, and uses lower semicontinuity to show that its limit attains the infimum.

A holomorphic or meromorphic integrand has a single-valued path integral on a domain whenever its integral around every closed curve in that domain vanishes.

If a cut joins two poles with opposite residues, every loop avoiding the cut has equal winding number about the endpoints, so the two residue contributions cancel.

Analytic continuation of an integral of a meromorphic function around a closed loop changes its value by $2\pi i$ times the winding-number-weighted sum of enclosed residues.

For steady incompressible flow with imposed outer pressure gradient,The normal momentum equation makes pressure approximately uniform across the layer.

Balancing $U^2/x$ with $\nu U/{\delta }^2$ gives $\delta \sim \sqrt{\nu x/U}$.

Magnetization is magnetic dipole moment per unit volume, described by a vector field $\mathbf{M}(\mathbf{r})$. Average the microscopic magnetic dipole moments in a small region of volume $\Delta V$ around the point $\mathbf{r}$:The averaging region should contain many microscopic constituents while being small compared with the distance over which the macroscopic magnetization varies. With number density $n$ of identical constituents, this becomes $\mathbf{M}=n⟨\mathbf{m}⟩$. Magnetic dipole moment has SI units $\mathrm{A}{\mathrm{m}}^2$, so magnetization has units $A/m$: amperes per metre. The average is a vector sum, not a sum of magnitudes.^[ref]

Alignment can change magnetization without changing the magnitude of any individual magnetic dipole moment. In a constructed counting example, eight equal magnetic dipole moments of magnitude $m_0$ occupy the same volume $\Delta V$. Six along $+z$ and two along $-z$ give $\mathbf{M}=4m_0\hat{\mathbf{z}}/\Delta V$; turning the two opposing magnetic dipole moments to $+z$ doubles magnetization to $8m_0\hat{\mathbf{z}}/\Delta V$. Both arrangements have the same sum of magnitudes, $8m_0$. Changing the individual magnetic dipole moments provides another possible response; their alignment and their magnitudes are distinct ingredients. Microscopic spin contributes magnetic dipole moment without being a literal rotating charged ball.

Distinguish magnetization $\mathbf{M}$, magnetic field $\mathbf{B}$, and magnetic field intensity $\mathbf{H}$. In SI,Here ${\mu }_0$ is the vacuum permeability, and $\mathrm{T}$ denotes the tesla. Magnetization describes matter's magnetic dipole moment density; magnetic field enters the Lorentz force; magnetic field intensity separates the free electric current contribution in the macroscopic equations. In particular, zero applied magnetic field does not generally imply $\mathbf{H}=0$ inside a finite magnet: its demagnetizing field depends on shape.^[ref]

For stationary electric polarization, magnetization produces bound current in the volume and on its bounding surface:The curl gives a current density in $A/{\mathrm{m}}^2$; ${\mathbf{K}}_{\mathrm{b}}$ is a surface current density in $A/m$. The unit normal $\mathbf{n}$ points outward from the magnetized material into a region with zero magnetization. Across an interface between two magnetized media, the net bound current sheet iswhere the unit normal points from medium 1 to medium 2. Both faces contribute: medium 2 has the opposite outward normal vector. This fixes the cross product sign. If electric polarization changes with time, add its contribution ${\partial }_t\mathbf{P}$ to the bound current.^[ref]

A short proof outline explains why both contributions are needed. Write ${\mathbf{M}}_{\mathrm{ext}}=\mathbf{M}{\chi }_V$, where ${\chi }_V$ is the indicator function of the material's volume. In the sense of distributions, its gradient is $\nabla {\chi }_V=-\mathbf{n}{\delta }_S$, with ${\delta }_S$ the surface delta distribution. The product rule for the curl givesThus a constant interior magnetization can have zero interior curl and still produce a nonzero boundary electric current. Equivalently, neighboring ideal microscopic current loops cancel along shared edges while uncompensated edges remain on the outside. This bound current representation predicts the same macroscopic magnetic field as the magnetization distribution; it does not require a wire or free charge transport around the magnet. Its usefulness is that the familiar Biot-Savart law can now calculate the magnetic field from a specified magnetization.

A time-dependent average magnetization need not vary smoothly under a smooth drive. The Barkhausen effect resolves collective jumps through their induced electromotive force; Barkhausen pulse area explains how that derivative readout relates to the underlying change.

The critical dimension is the target-spacetime dimension in which quantum anomalies obstructing Lorentz or Weyl symmetry cancel. It is 26 for the bosonic string.

Here critical means cancellation of the string’s quantum anomalies; it does not mean the upper or lower critical dimension of a phase transition.

The Fourier modes ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.

The Polyakov action introduces an independent worldsheet metric $g_{\alpha \beta }$:Its metric equation makes $g_{\alpha \beta }$ conformal to the induced worldsheet metric, reducing it classically to the Nambu–Goto action.

String tension is energy per unit length. The Regge-slope parameter ${\alpha }^{\prime }$ is conventionally defined by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$.

For a product state, probabilities of local measurement outcomes multiply.

For two labelled orthonormal bases, the event of equal labels is represented by ${\sum }_j|x_j{y}_j⟩⟨x_j{y}_j|$.

Starting the answer qubit in $|0⟩$, the sequencemaps $|x⟩|0⟩$ to $(-1{)}^{f(x)}|x⟩|0⟩$. It realizes a phase oracle with two standard-oracle queries while returning the answer register to its initial state.

The Bondi news is the retarded-time derivative of the leading asymptotic shear and measures outgoing gravitational radiation at null infinity.

Expanding an angular distribution in multipoles converts the Boltzmann equation into a hierarchy in which free streaming couples neighboring multipoles and collisions damp or source selected moments.