# Divergence

of a vector field $\mathbf{a}$ at a point $x = (x^{1},\ldots,x^{n})$

The scalar field $$x \mapsto \sum_{i = 1}^{n} \frac{\partial}{\partial x^{i}} [{a^{i}}(x)],$$ where the $a^{i}$’s are the components of the vector field $\mathbf{a}$.

The divergence of a vector field $\mathbf{a}$ at a point $x$ is denoted by $(\operatorname{div} \mathbf{a})(x)$ or by the inner product $\langle \nabla,\mathbf{a} \rangle (x)$ of the Hamilton operator $\nabla \stackrel{\text{df}}{=} \left( \dfrac{\partial}{\partial x^{1}},\ldots,\dfrac{\partial}{\partial x^{n}} \right)$ and the vector $\mathbf{a}(x)$.

If the vector field $\mathbf{a}$ is the field of velocities of a stationary flow of a non-compressible liquid, then $(\operatorname{div} \mathbf{a})(x)$ coincides with the intensity of the source (when $(\operatorname{div} \mathbf{a})(x) > 0$) or the sink (when $(\operatorname{div} \mathbf{a})(x) < 0$) at the point $x$.

The integral $$\int_{E} \operatorname{div}(\rho ~ \mathbf{a}) ~ \mathrm{d}{x},$$ where $\rho$ is the density of the liquid computed for the $n$-dimensional domain $E$, is equal to the amount of the liquid ‘issuing’ from $E$ per unit time. This amount (cf. Ostrogradski’s Formula) coincides with the magnitude $$\int_{\partial E} \langle \mathbf{N},\rho ~ \mathbf{a} \rangle ~ \mathrm{d}{S} = \sum_{i = 1}^{n} \int_{\partial E} N_{i} \rho a^{i} ~ \mathrm{d}{S},$$ where $\mathbf{N} = (N_{1},\ldots,N_{n})$ denotes the exterior unit normal vector to $\partial E$, and $\mathrm{d}{S}$ is the area element of $\partial E$. The divergence $(\operatorname{div} \mathbf{a})(x)$ is then the derivative with respect to the rate of the flow $\mathbf{a}$ across the closed boundary surface $\partial E$: $$(\operatorname{div} \mathbf{a})(x) = \lim_{E \to \{ x \}} \frac{1}{\operatorname{Vol}(E)} \int_{\partial E} \langle \mathbf{N},\mathbf{a} \rangle ~ \mathrm{d}{S}$$ Thus, the divergence is invariant with respect to the choice of coordinate system.

In curvilinear coordinates $y = (y^{1},\ldots,y^{n})$, we have $$(\operatorname{div} \mathbf{a})(y) = \frac{1}{\sqrt{g}} \sum_{i = 1}^{n} \frac{\partial}{\partial y^{i}} \left[ \sqrt{g} a^{i} \right], \quad \text{with} \quad g \stackrel{\text{df}}{=} \det([g_{ij}]) \quad \text{and} \quad g_{ij} \stackrel{\text{df}}{=} \sum_{\alpha = 1}^{n} \frac{\partial y^{\alpha}}{\partial x^{i}} \frac{\partial y^{\alpha}}{\partial x^{j}}, \qquad (\star)$$ where $\displaystyle \mathbf{a}(y) \stackrel{\text{df}}{=} \sum_{i = 1}^{n} {a^{i}}(y) ~ {\mathbf{s}_{i}}(y)$, and ${\mathbf{s}_{i}}(y)$ is the unit tangent vector to the $i$-th coordinate line at the point $y$: $${\mathbf{s}_{i}}(y) \stackrel{\text{df}}{=} \frac{1}{\sqrt{g_{ii}}} \frac{\partial y}{\partial x^{i}}.$$ The divergence of a tensor field $$x \mapsto a(x) = \left\{ {a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) ~ \middle| ~ i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n} \right\}$$ of type $(p,q)$, defined on an $n$-dimensional manifold with an affine connection, is defined with the aid of the corresponding absolute (covariant) derivatives of the components of $a(x)$, with subsequent convolution (contraction), and is a tensor of type $(p - 1,q)$ with components $${b^{i_{1} \ldots i_{s - 1} i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x) = \sum_{k = 1}^{n} {\nabla_{k} a^{i_{1} \ldots i_{s - 1} k i_{s + 1} \ldots i_{p}}_{j_{1} \ldots j_{q}}}(x), \qquad k,i_{\alpha},j_{\beta} \in \mathbb{N}_{\leq n}.$$ In tensor analysis and differential geometry, a differential operator operating on the space of differential forms and connected with the operator of exterior differentiation is also called a divergence.

#### References

 [1] N.E. Kochin, “Vector calculus and fundamentals of tensor calculus”, Moscow (1965). (In Russian) [2] P.K. [P.K. Rashevskii] Rashewski, “Riemannsche Geometrie und Tensoranalyse”, Deutsch. Verlag Wissenschaft. (1959). (Translated from Russian)

The Hamilton operator is usually called the nabla operator, after the symbol for it, $\nabla$. Ostrogradski’s formula is better known as the Gauss–Ostrogradski formula or the Gauss formula.
Let $M$ be an $n$-dimensional manifold and $\omega$ a volume element on $M$. The Lie derivative ${L_{X}}(\omega)$ is then also a differential $n$-form, and so ${L_{X}}(\omega) = f_{X} \omega$ for some function $f_{X}$ on $M$. This function is the divergence $\operatorname{div}(X)$ of $X$ with respect to the volume element $\omega$. If $g$ is a Riemannian metric on $M$, then the divergence of $X$ as defined by $(\star)$ above is the divergence of $X$ with respect to the volume element $\omega_{g} \stackrel{\text{df}}{=} \sqrt{\det(g)} \cdot \mathrm{d}{x^{1}} \wedge \cdots \wedge \mathrm{d}{x^{n}}$ defined by $g$. For any function $f$, note that $f \omega$ is an $n$-form, so $\displaystyle \int_{M} f \omega$ is defined — this is just the integral of $f$ with respect to the volume element $\omega$. If $M$ is compact, then Green’s theorem says that $$\int_{M} \operatorname{div}(X) ~ \omega = 0.$$ Still another notation for the divergence of an $n$-tuple $\mathbf{a} = (a^{1},\ldots,a^{n})$ of functions of $x_{1},\ldots,x_{n}$ (or of a vector field) is $\nabla \cdot \mathbf{a}$.