[Physics] Dot product and divergence

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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?

Best Answer

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 three-vectors, this expression for divergence is not commutative, as the $\nabla$ operator is not a vector in $\mathbb{R}^3$.