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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 boundary conditions. Its weak formulation is often an equation for a bounded bilinear form on a Sobolev space.

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.

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 $\Delta f=g^{\prime \prime }+(n-1)g^{\prime }/r$.

Spatial reflection reverses every spatial coordinate.

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 $\pm m$.

Operator ordering gives

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 $[X_i,P_j]=i\hslash {\delta }_{ij}$,