An algebraic simplification proves equality only where both original and transformed expressions are defined. If a cancelled factor vanishes at a point, the simplified expression may instead describe an extension to that point. For example,holds only when $\cos x\ne 0$ and $\cos x+\sin x\ne 0$. The right side is zero at $x=\pi /2+k\pi$, but the left side is undefined there. A graph of the original function must retain those holes. The extension may be useful, but it is a different function with a larger domain of a function.

On $0<x<L_x$, $0<y<L_y$ with positive side lengths and zero Dirichlet boundary conditions, separation of variables for $-\Delta u=\lambda u$ giveswith the displayed eigenvalues. The one-dimensional endpoint conditions exclude constant and hyperbolic factors and force the positive integer sine frequencies. The product Fourier sine series basis is complete in the square-integrable functions on the rectangle. Consequently an eigenfunction with prescribed $\lambda$ is a linear combination of exactly those modes with ${\lambda }_{mn}=\lambda$; if none match, only the zero solution exists. Distinct pairs with the same sum give an eigenvalue with geometric multiplicity greater than one. On a square of side $\pi$, for instance, $\lambda =25$ has precisely the pairs $(3,4)$ and $(4,3)$, so its eigenspace has dimension two.

For $\mu >0$, the decaying solution of $-y^{\prime \prime }+y=1_{[0,\mu ]}$ on the positive half-line with $y(0)=0$ follows by integrating the Dirichlet half-line Green function for minus d2 plus 1:Both $y$ and $y^{\prime }$ are continuous at $\mu$, whereas $y^{\prime \prime }$ has the jump required by the bounded step in the forcing. Its common value there is $(1-e^{-\mu }{)}^2/2$.

For the upper half of a three-dimensional ball, let $\overline{y}$ be the reflection of an interior source $y$ in the equatorial plane. The Dirichlet Green function inside a sphere, evaluated at the source and its reflection and then subtracted, vanishes on both the spherical cap and flat equatorial disc. Reflection makes the source distances and image distances agree on the plane. Only the original source is in the upper half-ball; the reflected source is in the lower half-ball, and both inversion images are outside the ball. Thus the difference has precisely the required Dirac delta distribution and Dirichlet boundary conditions.

The Dirac Hamiltonian is the first-order time-evolution operator obtained from the Dirac equation. For a free massive particle,where the Dirac matrices satisfy $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$, $\{{\alpha }_i,\beta \}=0$ and ${\beta }^2=I$. These identities give $H^2=c^2{P}^2+m^2{c}^4$, recovering the relativistic energy-momentum relation.

Minimal electromagnetic coupling replaces the canonical momentum by $\mathbf{P}-q\mathbf{A}$ and adds $q\varphi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The Dirac field interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.

For constant $A,\rho ,K$ on all space and time, the active waveform transformation ${\psi }_b(z,t)=\psi (bz,bt)$ leaves the action of $\mathcal{L}=A(\rho {\dot{\psi }}^2-K{\psi }^{\prime 2})/2$ invariant: the density gains $b^2$ and the integration measure gains $b^{-2}$. With canonical momentum $\Pi =A\rho \dot{\psi }$, its Noether current can be chosen asThe integrated charge $D=\int j_D^{0}dz$ is conserved if its boundary flux vanishes and the weighted integral exists. For a rightgoing wave packet $\psi =f(z-ct)$, $c=\sqrt{K/\rho }$, it becomes $D=-A\rho c\int \xi [f^{\prime }(\xi ){]}^{2}d\xi$. A fixed finite interval generally breaks this scale symmetry unless its boundaries and boundary data are transformed too.

For a first-derivative real scalar field in one space and one time dimension, put $j_D^{\mu }={\mathcal{L}}_{{\partial }_{\mu }\psi }(z{\psi }^{\prime }+t\dot{\psi })-x^{\mu }\mathcal{L}$. Its on-shell divergence isA quadratic derivative density with arbitrary field-dependent coefficients has zero derivative-homogeneity term. An extra term $-\alpha {\psi }^{\prime 4}$ instead gives $-2\alpha {\psi }^{\prime 4}$. A position-dependent kinetic coefficient $A\rho (z)/2$ gives $-Az{\rho }^{\prime }(z){\dot{\psi }}^2/2$. These source terms distinguish scale invariance from internal field-shift symmetry.

Use $D_n$ for the group of order $2n$ with rotation $R$ and reflection $m$. Every element is $R^j$ or $R^{j}m$, $0\le j<n$. The rotational conjugacy classes are the sets $\{R^j,R^{-j}\}$, reduced to singletons for central rotations. For odd $n$, all reflections form one class, giving $(n+3)/2$ classes in total. For even $n$, the identity and $R^{n/2}$ are central singletons, the other rotations form pairs, and the reflections split into even and odd powers of $R$, giving $n/2+3$ classes. Indeed conjugation by $R^s$ changes a reflection exponent by $2s$, while conjugation by $m$ negates it. For odd prime $n$, a normal subgroup containing a reflection must contain all reflections and hence the whole group; the proper normal subgroups are therefore only the identity subgroup and the rotation subgroup.

A missing diffraction order occurs when a principal maximum of an array factor coincides with a zero of the individual aperture's diffraction envelope. For identical slits of width $a$ and centre spacing $D$, order $m$ has $\sin {\theta }_m=m\lambda /D$. It is missing if $ma/D$ is a nonzero integer. This is cancellation of field contributions, not loss of the incident wavelength or a failure of the grating equation.

The obliquity factor weights the directional contribution of an aperture element in a scalar diffraction integral. In a fixed observation direction in the Fraunhofer diffraction regime, its variation across a sufficiently small aperture can be neglected along with the variation of the geometric amplitude denominator. Its angular dependence can still matter when comparing widely separated observation directions.

In 1992 the COBE DMR team, including George F. Smoot and Charles L. Bennett, used radiometer antennas to compare antenna temperatures between sky directions $60^{\circ }$ apart at $31.5$, $53$ and $90\mathrm{GHz}$, with two nearly independent channels per frequency. Repeated sky coverage supported least-squares solution map reconstruction.^[ref]

The analysis fitted magnetic response and removed the assumed kinematic angular dipole and its second-order Doppler effect contribution. Channel sums retained sky structure; channel differences estimated noise. Cross-channel correlations, different frequency maps and Milky Way latitude cuts tested the interpretation. Residual large-angle structure supported Cosmic microwave background anisotropy; Galactic contamination particularly limited the cosmic microwave background quadrupole. Brightness fluctuations alone were not sufficient evidence of primordial structure.^[ref]

The two differences serve different purposes. Comparing two sky directions measures angular variation. Comparing two independently calibrated receiver maps observing the same sky tests noise and consistency. A common sky signal cancels in the latter comparison if their antenna beam patterns and response scales match; a systematic error shared by both channels need not cancel. Separating observer motion, local emission and intrinsic Cosmic microwave background anisotropy requires these additional tests.

For an ideal Diesel cycle with compression ratio $r=V_1/V_2>1$, cutoff ratio $\rho =V_3/V_2>1$ and constant heat-capacity ratio $\gamma$, its thermal efficiency is $\eta =1-({\rho }^{\gamma }-1)/[\gamma r^{\gamma -1}(\rho -1)]$. As $\rho \to 1^{+}$ the quotient tends to $\gamma$, giving $\eta \to 1-r^{1-\gamma }$. The maximum-to-minimum temperature ratio tends to $r^{\gamma -1}$, so the limiting efficiency equals that of a Carnot cycle between those temperatures. Both heat input and net work vanish in this limit: the result is a limit of their ratio, not a finite-work engine at exactly zero cutoff interval.

Expand the determinant along any fixed row $i$ using the displayed formula, where $A^{(i,j)}$ deletes row $i$ and column $j$. The signed matrix minors are its cofactors. Applying the same formula to the transpose gives the column expansion. The row formula follows from the Leibniz formula for determinants by grouping permutations according to the selected column in row $i$. For a lower triangular matrix, expanding along the first row reduces the determinant to $A_{11}$ times a smaller triangular determinant; induction gives the product of all diagonal entries, even when some vanish.

Suppose $\psi$ is differentiable at zero, $\psi (0)=0$ and ${\psi }^{\prime }(0)=a\ne 0$. Since $\psi (x)/x\to a$, the displayed function satisfies $F(x)\sim a\mathrm{sgn}(x)|x{|}^{p+1}$ near zero. It is a continuous function at zero exactly when $p>-1$. Its difference quotient is $|h{|}^p\psi (h)/h$, so it has a finite derivative there exactly when $p\ge 0$: the value is $a$ for $p=0$ and zero for $p>0$. The nonzero-slope assumption is essential; a higher-order zero would change both thresholds.

Apply the chain rule to the logarithm of a differentiable nonzero real-valued function. This converts products into sums and powers into multiplicative coefficients. In particular, for positive $f=C{\prod }_i{x}_i^{a_i}$, $df/f={\sum }_i{a}_{i}d{x}_i/x_i$. It is useful for monomial uncertainty propagation and for finding relative rates of change.

For a real exponent $p$ and $x>0$, write $x^p=\exp (p\ln x)$. The chain rule and the derivative of the natural logarithm give $d(x^p)/dx=p{x}^{p-1}$. For a nonnegative integer exponent this agrees with polynomial differentiation and extends to all real $x$. Reversing differentiation gives $\int x^{p}dx=x^{p+1}/(p+1)+C$ for $p\ne -1$ on a suitable real branch. The exceptional exponent $p=-1$ gives $\ln |x|+C$ on either half-line.

Related rates use the chain rule to connect time derivatives of quantities constrained by an equation. If $F(x(t),y(t))=0$, differentiating gives $F_x\dot{x}+F_y\dot{y}=0$. For a rectangle with one fixed side $a$ and variable side $b(t)$, its diagonal $d(t)=\sqrt{a^2+b(t{)}^2}$ obeys $\dot{d}=(b/d)\dot{b}$. The equation must be differentiated before inserting an instantaneous value of $b$, since that value does not make the changing side constant in time.

For a homogeneous linear isotropic dielectric in vacuum, $P_i={ϵ}_0({ϵ}_r-1)E_{\mathrm{in},i}$. Combining this with the depolarization field and the imposed uniform electric field gives $E_{\mathrm{in},i}=E_{0,i}/[1+N_i({ϵ}_r-1)]$. A sphere has $N_i=1/3$. An infinite slab has $N=1$ normal to its faces and $N=0$ in its plane. A very long circular rod has $N=0$ along its axis and $N=1/2$ transverse to it. Slab and rod statements are limiting geometries, with end effects neglected; a finite cylinder does not generally have an exactly uniform internal electric field.

In deep inelastic scattering, the reference frame where the exchanged virtual photon has zero energy and spatial momentum magnitude $Q$.

Starting in 1967, a collaboration at the Stanford Linear Accelerator Center and the Massachusetts Institute of Technology directed an electron beam at liquid hydrogen, providing proton targets. Jerome Isaac Friedman, Henry Way Kendall and Richard Edward Taylor participated in this experimental program.^[ref]

The 1969 paper Observed Behavior of Highly Inelastic Electron-Proton Scattering reports inclusive electron spectra at $6^{\circ }$ and $10^{\circ }$ with incident energies from $7$ to $17\mathrm{GeV}$. Its scaling comparison estimated ${\nu }_{\mathrm{lab}}{W}_2$ under a transverse-virtual photon assumption, finding approximate agreement as a function of $\omega =1/x$ for $Q^2>1{\mathrm{GeV}}^2$ and $\omega >4$. These are the paper's conditions, not a claim of exact Bjorken scaling at every momentum transfer.^[ref]