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For a linear map $L:V\to W$ and one solution $L{u}_0=f$, all solutions are $u_0+\ker L$. This follows because $L(u-u_0)=0$, and conversely adding any kernel element preserves the equation. If $U\subseteq V$ is a vector subspace representing a desired regularity class and $u_0\in U$, then every solution to this one inhomogeneous equation lies in $U$ exactly when $\ker L\subseteq U$. For distributional differential equations, take $U$ to be the distributions represented by smooth functions.

A bump function is a smooth function with compact support. Rescaled and translated bump functions provide local cutoffs.

The Lie derivative of a tensor field satisfies $[{\mathcal{L}}_X,{\mathcal{L}}_Y]={\mathcal{L}}_{[X,Y]}$. On smooth functions this is the definition of the Lie bracket of vector fields; on vector fields it follows from the Jacobi identity. The Leibniz rule and contraction compatibility extend the equality to tensor fields. The cyclic double-commutator identity also follows directly from associativity of operator composition.

If the constant symmetric matrix $A$ satisfies $A\xi \cdot \xi \ge \theta |\xi {|}^2$, then every smooth function $u$ with compact support satisfiesIndeed, the Fourier transform of a derivative and the Plancherel theorem turn the squared norms into integrals with Fourier multipliers $|\xi {|}^4$ and $(A\xi \cdot \xi {)}^2$.

For $T\in {\mathcal{S}}^{\prime }$ and $\psi \in \mathcal{S}$, define $(T\ast \psi )(x)=⟨T_y,\psi (x-y)⟩$. Translations are smooth in the Schwartz space, so every derivative is ${\partial }^{\alpha }(T\ast \psi )(x)=⟨T,{\partial }^{\alpha }\psi (x-\cdot )⟩$. The finite-seminorm continuity estimate for $T$ implies $|{\partial }^{\alpha }(T\ast \psi )(x)|\le C_{\alpha }(1+|x|{)}^m$ for one $m$. Thus the result is a smooth function defining a tempered distribution, but it need not be a Schwartz function: $1\ast \psi =1$ when $\int \psi =1$. If both factors are radial, testing the rotation action on the translated $\psi$ shows that the convolution is radial.

The difference of two affine connections is a smooth tensor field of type $(1,2)$. Linearity over smooth functions holds in the first slot by the connection axioms. In the second slot, the two extra terms $X(f)Y$ cancel, so $\Delta (X,fY)=f\Delta (X,Y)$. This tensoriality makes the value depend only on $X_p,Y_p$. In coordinates its components are the differences of the Christoffel symbols, although either collection of connection coefficients individually is not a tensor.

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

A smooth function on a real open interval is real analytic exactly when, near each point, there are constants $C,A>0$ such that $|f^{(n)}(x)|\le C{A}^{n}n!$ for every $n$ and every $x$ in that neighborhood. The bound must hold on a neighborhood, not only at its center. The Taylor theorem with Lagrange remainder proves sufficiency; differentiating a convergent power series on a smaller interval proves necessity. A flat function shows why pointwise bounds alone are insufficient.

A sheaf of modules over the sheaf of smooth functions is fine when every locally finite open cover admits a subordinate family of sheaf endomorphisms summing to the identity. Fine sheaves on a paracompact space are acyclic, so their positive-degree sheaf cohomology vanishes.

A smooth function is flat at $a$ when every derivative, including order zero, vanishes at $a$. The function $e^{-1/x^2}$ off zero, extended by zero at zero, is flat there but not identically zero near zero. Thus its Taylor series does not determine it. A real analytic function that is flat at a point vanishes on a neighborhood of that point.

For $u\in {\mathcal{E}}^{\prime }({\mathbb{R}}^n)$, its Fourier transform is the smooth functionIt has at most polynomial growth, and its extension to complex frequency is controlled more precisely by the Paley–Wiener–Schwartz theorem.

In three spatial dimensions, with $Z$ the eleven translations, spatial rotations, boosts and scaling commutation vector fields for the wave equation, a sufficiently decaying smooth function satisfies $|u(t,x)|\le C(1+t+|x|{)}^{-1}(1+|t-|x||{)}^{-1/2}{\sum }_{|I|\le 2}\Vert Z^{I}u(t){\Vert }_2$. This weighted Sobolev inequality converts control of commuted wave energy into decay. It holds independently of any wave equation satisfied by $u$.

For a smooth function $f$ and a vector field $X$, the Lie derivative is ${\mathcal{L}}_{X}f=X(f)=df(X)$. It is the derivative of $f$ along the local flow of $X$.

With no axial electric current, cylindrical magnetostatic pressure balance reduces to constant $p+B_z^2/(8\pi )$. For $p=p_0{e}^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions areAs $R\downarrow 0$, $B_z\sim \sigma \sqrt{8\pi p_0}R/a$ and $j_{\varphi }\to -\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a smooth function of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.

Mollification is the regularization of a function or distribution by convolution with a rescaled mollifier. It creates smooth functions on interior regions, preserves constant-coefficient distributional equations there, and converges back to the original object in the appropriate local topology as the radius tends to zero. The process is distinct from the kernel used to perform it.

For a distribution $u$ and a smooth function $a$, the product $au$ is defined by $⟨au,\phi ⟩=⟨u,a\phi ⟩$. It obeys the Leibniz rule; in one dimension, $(au{)}^{\prime }=a^{\prime }u+a{u}^{\prime }$.

A compactly supported distribution of order of a distribution at most $M$ has a smooth function as its Fourier transform, bounded by $C⟨\xi {⟩}^M$. It therefore belongs to every Sobolev space $H^s$ with $s<-M-n/2$.

A stationary point $x_0$ of a smooth phase $\varphi$ is nondegenerate when its Hessian matrix at $x_0$ is an invertible matrix. In one dimension, this condition is ${\varphi }^{\prime \prime }(x_0)\ne 0$.