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For a $C^1$ flux $F$ and a weak solution of $\mathrm{div}F(Du)=0$, discrete integration by parts and the fundamental theorem of calculus along a line segment show that $w_h={\delta }_{\ell ,-h}u$ solves $\mathrm{div}(A_{h}D{w}_h)=0$, with $p_{-}=Du(x-h{e}_{\ell })$ and $p_{+}=Du(x)$. Coercive quadratic-form bounds and operator norm bounds for $DF$ on all segments between these gradients pass to $A_h$. This produces a divergence-form elliptic operator without initially assuming second partial derivatives of $u$.

Local coordinates $x^{\mu }$ induce the coordinate-basis tangent vectors ${\partial }_{\mu }=\partial /\partial x^{\mu }$. Under a coordinate change they transform by the chain rule.

Differentiability is the existence of a derivative at a point. For maps between finite-dimensional normed vector spaces, the existence of a Frechet derivative requires a single linear map approximating the increment with error $o(\Vert h\Vert )$. Merely having partial derivatives need not provide this approximation; continuous partial derivatives on a neighbourhood do suffice.

With $D={\partial }_{\mu }{A}^{\mu }$, expand the Maxwell Lagrangian and its Feynman gauge term to findCommuting partial derivatives verifies the divergence identity. Both densities give $\square A^{\mu }=0$ under the usual variational boundary conditions. Their canonical momentum components differ: the original density gives $-F^{0\nu }-{\eta }^{0\nu }D$, whereas the subtracted density gives $-{\dot{A}}^{\nu }$. Thus a free-photon momentum expansion using the latter requires a stated boundary-term convention.

A linear partial differential equation is a partial differential equation in which the unknown function and its partial derivatives occur linearly, with coefficients independent of the unknown. Its homogeneous solutions obey the superposition principle.

A mixed partial derivative takes partial derivatives in different variables, for example $f_{xy}={\partial }_y({\partial }_{x}f)$. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged: $f_{xy}=f_{yx}$. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

For a Newtonian gravitational potential obeying the Poisson equation for Newtonian gravity, define $T_{ij}=[{\partial }_i\Phi {\partial }_j\Phi -{\delta }_{ij}|\nabla \Phi {|}^2/2]/(4\pi G)$. Commutation of mixed partial derivatives gives ${\partial }_j{T}_{ij}=\rho {\partial }_i\Phi$. Thus the gravitational force density can be put in conservative form as $-{\partial }_j{T}_{ij}$. The sign is chosen so that $T_{ij}$ is added to the momentum flux; reversing the stress convention reverses the divergence sign.