Divergence and curl of a $C^1$ vector field

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and let $F=(F_0,\dots ,F_{n-1}):U\to {\mathbb{R}}^n$ be $C^1$ in the componentwise Euclidean sense of $C^k$ Euclidean maps and diffeomorphisms. Then the divergence of $F$ is $\mathrm{div}F:={\sum }_{i<n}{\mathrm{\partial }}_i{F}_i$, the function $U\to \mathbb{R}$ whose value at $p$ is ${\sum }_{i<n}{\mathrm{\partial }}_i{F}_i(p)$. The partial derivatives are those of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and the sum is the finite sum used throughout The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Since each ${\mathrm{\partial }}_i{F}_i$ is continuous on $U$, so is $\mathrm{div}F$.

Now let $n=3$ and let $F:U\to {\mathbb{R}}^3$ be $C^1$ on an open $U\subseteq {\mathbb{R}}^3$. Following The cross product in ${\mathbb{R}}^3$, write the three coordinates of a point and of a vector as $x,y,z$ rather than $0,1,2$, so that $F=(F_x,F_y,F_z)$ means $F=(F_0,F_1,F_2)$ and ${\mathrm{\partial }}_x,{\mathrm{\partial }}_y,{\mathrm{\partial }}_z$ are ${\mathrm{\partial }}_0,{\mathrm{\partial }}_1,{\mathrm{\partial }}_2$. With that naming, the curl of $F$ is $\mathrm{curl}F:=({\mathrm{\partial }}_y{F}_z-{\mathrm{\partial }}_z{F}_y,{\mathrm{\partial }}_z{F}_x-{\mathrm{\partial }}_x{F}_z,{\mathrm{\partial }}_x{F}_y-{\mathrm{\partial }}_y{F}_x)$, a map $U\to {\mathbb{R}}^3$ each of whose coordinates is continuous on $U$. In this naming the divergence reads $\mathrm{div}F={\mathrm{\partial }}_x{F}_x+{\mathrm{\partial }}_y{F}_y+{\mathrm{\partial }}_z{F}_z$.

Both operators are defined pointwise from the first partial derivatives of the components, so no differentiability of $F$ beyond $C^1$ is used and no orientation or metric structure enters beyond the standard coordinates of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. For a $C^1$ scalar function $f$ on $U$, the gradient $\mathrm{\nabla }f=({\mathrm{\partial }}_{0}f,\dots ,{\mathrm{\partial }}_{n-1}f)$ is that of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case; in the three-coordinate naming, $\mathrm{\nabla }f=({\mathrm{\partial }}_{x}f,{\mathrm{\partial }}_{y}f,{\mathrm{\partial }}_{z}f)$.

• Why the curl is only defined in three coordinates. If $A=JF-(JF{)}^{\mathsf{T}}$ with the row-component Jacobian convention, then the curl coordinates are $A_{zy}$, $A_{xz}$ and $A_{yx}$. Thus the curl is encoded, with fixed signs, by the three independent off-diagonal entries of $A$ (or by twice those entries if “antisymmetric part” means $A\mathrm{/}2$). In $n$ coordinates there are $n(n-1)\mathrm{/}2$ independent entries. Only at $n=3$ is that number again $n$, which is what allows the collection to be read as a vector in the same space. The divergence has no such restriction and is defined for every $n\ge 1$.

• The word "divergence" here is about vector fields. It has nothing to do with the divergence of a sequence or of a series; the two senses share only the word.

• Placement of the minus sign in the second coordinate. Some presentations write the middle coordinate as $-({\mathrm{\partial }}_x{F}_z-{\mathrm{\partial }}_z{F}_x)$. That is the same real number as ${\mathrm{\partial }}_z{F}_x-{\mathrm{\partial }}_x{F}_z$, and the form displayed above is the one whose three coordinates read off the coordinate formula of The cross product in ${\mathbb{R}}^3$ in the same cyclic pattern.