Dot product and divergence
It is pretty much simply a short way to notate both vector field operations by looking at $\nabla$ as a vector operator by writing \begin{equation} \nabla=\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right) \end{equation} in $\mathbb{R}^3$, or equivalently \begin{equation} \nabla=\frac{\partial}{\partial x}\hat{\imath}+\frac{\partial}{\partial y}\hat{\jmath}+\frac{\partial}{\partial z}\hat{k}. \end{equation} Performing this vector operator on a scalar field gives you the expression for that field's gradient, whereas applying it to a vector field via a dot product gives you the vector field's divergence (analogoulsy for the cross product, which gives you the field's curl instead).
It is important to note, however, that unlike with regular threevectors, this expression for divergence is not commutative, as the $\nabla$ operator is not a vector in $\mathbb{R}^3$.
nelson ningombam
Updated on January 24, 2020Comments

nelson ningombam almost 4 years
Divergence is represented by dot product. How is the divergence related to dot product? And curl is represented by cross product. How is the curl related to cross product?

Admin almost 8 years$\operatorname{div} F = \nabla \cdot F$ , $\operatorname{curl} F = \nabla \times F$

Suriya almost 8 yearsIts just the dot/cross product of the Del operator with the vector

ACuriousMind almost 8 yearsI'm voting to close this question as offtopic because it's not a physics question and I consider it of too low quality to migrate it.

dmckee  exmoderator kitten almost 8 yearsThinking about the meaning of these notations (and that for the gradient) will start you on the road to understanding the algebra of differential operators.

nelson ningombam almost 8 yearsit is related to physics...without this knowledge how 1 can understand the maxwell`s equations @ACuriousMind


nelson ningombam almost 8 yearsi know that the dot product represents the divergence, but why the dot product represents the divergence?