# 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.

How to Cite This Entry:
Divergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence&oldid=39958
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article