Curl and Divergence

Vector Fields

A vector functions of three variables is defined as

(1)   \begin{equation*}\boldsymbol{F} (x,y,z) = P(x,y,z) \boldsymbol{i} + Q(x,y,z) \boldsymbol{j} + R(x,y,z) \boldsymbol{k} .\end{equation*}

Curl of a Vector Field

The curl of a vector field is a vector field itself, defined as

(2)   \begin{equation*}curl \boldsymbol{F} = \boldsymbol{\nabla} \times \boldsymbol{F} .\end{equation*}


Watch the following YouTube video for 2D curl intuition.

Divergence of a Vector Field

The divergence of a vector field is a scalar defined as

(3)   \begin{equation*}div \boldsymbol{F} = \boldsymbol{\nabla} \cdot \boldsymbol{F} .\end{equation*}


Watch the following YouTube videos for divergence intuition.

Reference

Dennis G. Zill. Advanced Engineering Mathematics, 6^{th} edition. Jones & Bartlett Learning. 2016.