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A tensor derivation is a real-linear operation preserving every tensor type, obeying the tensor-product Leibniz rule, and commuting with every tensor contraction. A derivation $D$ of smooth functions and a real-linear operator on vector fields satisfying $D(fY)=fDY+(Df)Y$ extend uniquely to a tensor derivation. On a differential one-form it must satisfyThis expression is linear over smooth functions in $Y$. In a local frame $D{e}_a=A^b{}_a{e}_b$, the dual rule is $D{ϵ}^a=-A^a{}_b{ϵ}^b$. Apply the product rule to every coefficient and frame factor to define the extension; these dual signs cancel under contraction. Frame changes agree by differentiating the inverse matrix. Cutoffs prove locality and hence uniqueness from local expansions.

A multilinear operation on vector fields is tensorial when it is linear over smooth functions in every argument. Its value at a point then depends only on the argument vectors at that point. To see this, use a smooth cutoff function to reduce to local fields, expand them in a local frame, and apply linearity over smooth functions to their coefficients. A smoothly valued tensorial operation therefore defines a tensor field. The curvature of an affine connection is tensorial, whereas a covariant derivative differentiates a scalar coefficient in its second argument and is not tensorial there.

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set $K$. Each ${\mathcal{D}}_K$ uses seminorms $p_m(\phi )={\max }_{|\alpha |\le m}\Vert {\partial }^{\alpha }\phi {\Vert }_{\infty }$. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each ${\mathcal{D}}_K$ satisfies a finite order of a distribution estimate; the order and constant may depend on $K$.

For the constant-velocity linear transport equation $u_t+c\cdot {\nabla }_{x}u=0$, a Dirac delta distribution initially at zero moves along $x=ct$. Its spacetime distribution pairs by $⟨u,\phi ⟩=\int \phi (ct,t)dt$, and its initial trace is obtained by weak convergence of distributions. The Fourier representation of the Dirac delta function writes it as an oscillatory integral with phase function $(x-ct)\cdot \theta$. Its singular support is precisely the trajectory; it has order-zero distribution regularity as a measure, without becoming a smooth function.