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)

Comments

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.

For other vector differentiation operators, see Curl; Gradient. For relations between these, see also Vector analysis.

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} $.

References

[a1]	D.E. Bourne, P.C. Kendall, “Vector analysis and Cartesian tensors”, Nelson & Sons, Sunbury-on-Thames (1977). Zbl 0781.53013