The master formula can be used to derive formulas for the gradient in other coordinate systems. We illustrate the method for polar coordinates.

In polar coordinates, we have $$ df = \Partial{f}{r}\,dr + \Partial{f}{\phi}\,d\phi $$ and of course $$ d\rr = dr\,\rhat + r\,d\phi\,\phat $$ which is (1) of § {Other Coordinate Systems}. Comparing these expressions with the Master Formula (4) of § {Gradient}, we see immediately that we must have \begin{equation} \grad f = \Partial{f}{r}\,\rhat + {{1}\over{r}}\Partial{f}{\phi}\,\phat \label{gradpolar} \end{equation} Note the factor of ${{1}\over{r}}$, which is needed to compensate for the factor of $r$ in (1) of § {Other Coordinate Systems}. Such factors are typical for the component expressions of vector derivatives in curvilinear coordinates.

Why would one want to compute the gradient in polar coordinates? Consider the computation of $\grad\,\left({\ln\sqrt{x^2+y^2}}\right)$, which can done by brute force in rectangular coordinates; the calculation is straightforward but messy, even if you first use the properties of logarithms to remove the square root. Alternatively, using ($\ref{gradpolar}$), it follows immediately that $$ \grad\,\left({\ln\sqrt{x^2+y^2}}\right) = \grad\,({\ln r}) = {1\over r}\,\rhat $$

Exactly the same construction can be used to find the gradient in other coordinate systems. For instance, in cylindrical coordinates we have \begin{eqnarray*} dV = \Partial{V}{r} \,dr + \Partial{V}{y} \,d\phi + \Partial{V}{z} \, dz \end{eqnarray*} and since in cylindrical coordinates \begin{eqnarray*} d\rr = dr\,\rhat + r\,d\phi\,\phat + dz\,\zhat \end{eqnarray*} we obtain \begin{eqnarray*} \grad V = \Partial{V}{r} \,\rhat + \frac{1}{r} \Partial{V}{\phi} \,\phat + \Partial{V}{z} \,\zhat \end{eqnarray*} This formula, as well as similar formulas for other vector derivatives in rectangular, cylindrical, and spherical coordinates, are sufficiently important to the study of electromagnetism that they can, for instance, be found on the inside front cover of Griffiths' textbook, Introduction to Electrodynamics, and are also given in § {Formulas for Div, Grad, Curl}.