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For $r=|x|>0$, the gradient satisfies ${\partial }_{i}r=x_i/r$. The product rule and Einstein notation then give the displayed divergence formula in dimension $n$. In dimension three, its derivative matrix is symmetric, so the curl vanishes. Zero divergence forces $f(r)=C{r}^{-n}$ on each positive radial interval. This formula does not assert regularity at the origin.

For continuously differentiable vector fields, the gradient of their dot product expands as displayed. The contraction of two Levi-Civita symbols gives $[F\times (\nabla \times G){]}_i=F_j{\partial }_i{G}_j-F_j{\partial }_j{G}_i$; adding the directional derivative cancels the second term. Interchanging the fields and using the product rule proves the identity. It does not require either field to be irrotational.

Assume the scaling function Fourier transform and the MRA low-pass filter are $C^{p-1}$, and $m^{(k)}(\pi )=0$ for $k<p$. For every nonzero integer $j$, write $j=2^r\ell$ with $\ell$ odd. Iterating the scaling refinement equation at $2\pi j+t$ supplies a factor $m(\pi \ell +t/2^{r+1})$. All its derivatives of order less than $p$ vanish at $t=0$, by periodicity. The product rule gives ${\hat{\phi }}^{(k)}(2\pi j)=0$ for $k<p$.

For continuous semimartingales,It is the stochastic counterpart of the ordinary product rule; the quadratic covariation supplies the additional second-order term.

For a smooth map $u:{\mathbb{R}}^3\to {\mathbb{R}}^3$, set $V_i={ϵ}_{ijk}{ϵ}_{abc}({\partial }_j{u}_a)({\partial }_k{u}_b)u_c/6$. In Einstein summation convention, applying the product rule leaves only the term differentiating $u_c$: the terms with second derivatives cancel against antisymmetry. The remaining Levi-Civita symbol contraction is $\det Du$, so $\nabla \cdot V=J[u]$. Equivalently this vector field is one third of the cofactor matrix transposed, applied to $u$.