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A nonlinear partial differential equation depends nonlinearly on the unknown function or one of its derivatives.

The modified Helmholtz equation isIt reduces to the Laplace equation when $m=0$.

An elliptic boundary value problem couples an elliptic differential equation in a domain to specified boundary conditions. An important example is the Poisson equationHere the Dirichlet boundary data fix the boundary value. Prescribing outward flux instead gives a Neumann boundary-value problem, which has different compatibility and normalization requirements. Thus the differential expression alone does not specify the problem.

For zero Dirichlet boundary conditions, multiply the Poisson equation by a smooth compactly supported test function and apply integration by parts:This weak formulation requires only first weak derivatives, so it makes sense for $u\in H_0^1(\Omega )$ and $f\in L^2(\Omega )$. On a bounded domain, the Poincaré inequality makes its bilinear form coercive, and the Lax-Milgram theorem gives a unique weak solution. The right side also extends to suitable forcing in the continuous dual space of $H_0^1(\Omega )$.^[ref]

The energy method explains the value of this formulation: subtracting two solutions and testing with their difference proves uniqueness. Elliptic regularity asks whether additional smoothness of coefficients, forcing and boundary upgrades the weak solution to a classical one. A Schauder estimate quantifies this gain in Hölder spaces; rough coefficients instead lead to results such as the De Giorgi-Nash-Moser theorem.

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve. With constant speed $c$, the chain rule gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c{u}_x=0$, hence $u(x,t)=g(x-ct)$. This is exact translation: it preserves the shape rather than smoothing it, as the periodic heat and transport comparison shows.

For a spatially varying velocity field, distinguish an advected value from a conserved mass density. The continuity equation is ${\rho }_t+\mathrm{div}(b\rho )=0$, which expands to ${\rho }_t+b\cdot \nabla \rho +(\mathrm{div}b)\rho =0$. Thus along the characteristic flow map $X_t$, it is $\rho (t,X_t(\alpha ))\det D{X}_t(\alpha )$ that remains constant. For $b(x)=x$ on the real line, scalar transport equation gives $u(t,x)=g(e^{-t}x)$, whereas the continuity equation gives $\rho (t,x)=e^{-t}g(e^{-t}x)$. For integrable $g$, change of variables yields $\int u(t,x)dx=e^t\int g$ and $\int \rho (t,x)dx=\int g$. The expanding flow preserves carried values and spreads conserved mass over a growing volume.

A solution of a partial differential equation is a function satisfying the equation in a specified sense and obeying its initial or boundary data.

A quasilinear partial differential equation is linear in its highest-order derivatives, while its coefficients may depend on the independent variables, the unknown function, and lower-order derivatives.

An energy estimate tests a differential equation against the solution or one of its derivatives to control a norm by initial data and forcing. Coercive terms represent dissipation, while Young inequality and the Gronwall inequality control lower-order terms.

For the difference $w$ of two smooth heat equation solutions with $D>0$, identical PDE source term, initial conditions, and either periodic boundary conditions or Dirichlet boundary data, integration by parts givesThe initially zero nonnegative integral stays zero, proving uniqueness. The same computation bounds changes in the L2 norm when the initial conditions differ. Specifying data and a solution class makes this argument meaningful.^[ref]

A periodic domain identifies opposite faces of a fundamental cell, producing a flat torus. On an interval $[0,L]$, smooth periodic boundary conditions require $u(0,t)=u(L,t)$ and matching spatial derivatives. These endpoints are one location, rather than two independently controlled walls. Periodic integration by parts has no boundary term, and zero-mean functions have no zero Fourier mode.

A radial function depends only on the distance from the origin: $f(x)=g(|x|)$. Differential operators then reduce to radial formulas such as $\Delta f=g^{\prime \prime }+(n-1)g^{\prime }/r$.

Here spatial reflection means inversion of all three spatial coordinates, $\mathbf{r}\mapsto -\mathbf{r}$, as used in the parity operator. This differs geometrically from reflection in a single plane, which reverses only the perpendicular coordinate. In three dimensions inversion reverses orientation; in two dimensions simultaneous reversal of both coordinates is instead a half-turn.

Under three-dimensional inversion, a position or momentum vector changes sign, while angular momentum $\mathbf{L}=\mathbf{r}\times \mathbf{p}$ does not. This distinction explains why angular momentum is an axial vector. In quantum mechanics, inversion tests the parity of a wavefunction and whether a Hamiltonian exhibits parity conservation.

If $k\ge 1$ and $k+2\le p\le 2k+1$ is prime, thenIndeed, neither $(k+1)!$ nor $k!$ contains a factor $p$, while $(2k+1)!$ contains $p$ exactly once because $2p>2k+1$.

The 2-adic valuation is the p-adic valuation at the prime number $p=2$.

The dual coframe obeys ${\theta }^a(e_b)={\delta }_b^a$ and reconstructs the metric as $g={\eta }_{ab}{\theta }^a\otimes {\theta }^b$.

Forboth $P^2$ and $|X{|}^2$ are rotational scalars, so $[H,L_i]=0$. Consequently $H,L^2,L_3$ commute pairwise and can be simultaneously diagonalized.

The coordinate combinations $x_1\pm i{x}_2$ have $L_3$ eigenvalues $\pm \hslash$, so their powers carry magnetic quantum numbers $\pm m$.

The square of orbital angular momentum contains a correction to the classical identity because position operators and momentum operators do not commute. Work in three dimensions with $[X_i,P_j]=i\hslash {\delta }_{ij}$ on a common invariant domain, such as Schwartz space.

Operator ordering gives

Using $L_i={\epsilon }_{ijk}{X}_j{P}_k$ and contracting the two Levi-Civita symbols gives $L^2={\sum }_{j,k}(X_j{P}_k{X}_j{P}_k-X_j{P}_k{X}_k{P}_j)$. The first sum is $X^2{P}^2-i\hslash X\cdot P$ and the second is $(X\cdot P{)}^2-2i\hslash X\cdot P$, yielding the displayed result. Keeping the order is essential: replacing both sums by commuting products would discard the quantum term.

The operators $L_{\pm }=L_x\pm i{L}_y$ satisfyThey therefore change the magnetic quantum number $m$ by $\pm 1$ while preserving the total angular-momentum quantum number $\ell$.

The canonical position-momentum commutators imply

For $L_i={\epsilon }_{ik\ell }{X}_k{P}_{\ell }$ and $[X_i,P_j]=i\hslash {\delta }_{ij}$,

Weighting each configuration by $e^{\tau l}$ gives