Dot product and divergence

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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 $\mathbb{R}^3$.

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nelson ningombam
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nelson ningombam

Updated on January 24, 2020

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  • nelson ningombam
    nelson ningombam almost 3 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
      Admin almost 7 years
      $\operatorname{div} F = \nabla \cdot F$ , $\operatorname{curl} F = \nabla \times F$
    • Suriya
      Suriya almost 7 years
      Its just the dot/cross product of the Del operator with the vector
    • ACuriousMind
      ACuriousMind almost 7 years
      I'm voting to close this question as off-topic because it's not a physics question and I consider it of too low quality to migrate it.
    • dmckee --- ex-moderator kitten
      dmckee --- ex-moderator kitten almost 7 years
      Thinking 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
      nelson ningombam almost 7 years
      it is related to physics...without this knowledge how 1 can understand the maxwell`s equations @ACuriousMind
  • nelson ningombam
    nelson ningombam almost 7 years
    i know that the dot product represents the divergence, but why the dot product represents the divergence?