Let \( F : \Omega \to \mathbb{R}^d \) be differentiable. Then
\[ \operatorname{div} F = \sum_{i=1}^{d} \partial_i F_i. \]Rules Link to heading
It is, for \( F, G : \Omega \to \mathbb{R}^d \) differentiable and \( \alpha \in \mathbb{R} \),
\[ \operatorname{div}(\alpha F + G) = \alpha \, \operatorname{div} F + \operatorname{div} G. \]For a scalar-valued function \( \varphi \) and a vector field \( F \), the following product rule holds:
\[ \operatorname{div}(\varphi F) = \varphi \, \operatorname{div} F + \nabla \varphi \cdot F. \]