Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function
  Recalling that the operator $\nabla$ is given by
\begin{displaymath}\nabla = \hat\imath\frac{\partial F_x}{\partial x}+\hat\jmath... ...partial F_y}{\partial y}+\hat k \frac{\partial F_z}{\partial z}\end{displaymath}
  and using , we can write ${\rm div}\vec F= \nabla\cdot\vec F$.
  The following facts may be noted :
1.
 The divergence of a vector field is a scalar
2.
 $\nabla\cdot(\vec F+\vec G) = \nabla\cdot\vec F + \nabla\cdot\vec G$
3.
 $\nabla\cdot(\phi\vec F)= \phi\nabla\cdot\vec F + \nabla\phi\cdot\vec F$
4.
 In cylindrical coordinates
 
\begin{displaymath}\nabla\cdot\vec F = \frac{1}{\rho}\frac{\partial}{\partial\rh... ...tial}{\partial\theta}F_\theta + \frac{\partial} {\partial z}F_z\end{displaymath}
5.
 In spherical polar coordinates
 
\begin{displaymath}\nabla\cdot\vec F = \frac{1}{r^2}\frac{\partial}{\partial r}(... ...ta) + \frac{1}{r\sin\theta}\frac{\partial}{\partial\phi}F_\phi\end{displaymath}
6.
 The divergence theorem is
 
\begin{displaymath}\oint_S \vec F\cdot\vec{dS}= \int_V\nabla\cdot\vec F dV\end{displaymath}
 
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