Identity 5: $\nabla ·\left(f \mathbf{F}\right)\= f \nabla ·\mathbf{F}\+\mathbf{F}·\left(\nabla f\right)$

The left side is the divergence of the product $f \mathbf{F}$, where $f$ is a scalar and F is a vector, both in polar coordinates. Hence, the resulting scalar is

$\frac{\left(f u\right)}{r}\+{\partial }_r\left(f u\right)\+\frac{{\partial }_{\mathrm{\θ}}\left(f v\right)}{r}$ = $\frac{f u}{r}\+f_r u\+f u_r\+\frac{f_{\mathrm{\θ}} v\+f v_{\mathrm{\θ}}}{r}$ 

The first term on the right is the product of the scalar $f$ and the divergence of F. The resulting scalar is

$f\left(\frac{u}{r}\+u_r\+\frac{v_{\mathrm{\θ}}}{r}\right)\=\frac{f u}{r}\+f u_r\+\frac{f v_{\mathrm{\θ}}}{r}$ 

The second term on the right is the dot product of F with the gradient of $f$. The resulting scalar is

$\left[\begin{array}{c}u\\ v\end{array}\right]·\left[\begin{array}{c}{f}_r\\ \frac{f_{\mathrm{\θ}}}{r}\end{array}\right]$ = $u f_r\+\frac{v f_{\mathrm{\θ}}}{r}$ 

The sum of the two terms on the right is then

On the right, explicit use must be made of the Gradient command because $f$, a scalar, does not carry its coordinate system as an attribute. In typeset notation, the Del operator has no way of knowing which coordinate system to use when operating on a scalar. The alternative to invoking the Gradient command would be to set the ambient coordinate system to polar coordinates with the SetCoordinates command, an approach this Study Guide consistently avoids implementing.