You can nest align in gather (not the other way around). In the example code I use numbered equations but adding the * will produce the expected result.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
The the basic solution consists in nesting \texttt{align} in \texttt{gather}
as shown below
\begin{gather}
\begin{align}
r &= \sqrt{x^2 + y^2} \\
f &= m a \\
A &= \frac{\pi}{4} d^2
\end{align}\\
\frac{\partial P}{\partial \theta} = \frac{P_{(i+1,j)} - P_{(i-1,j)}}{2 \Delta\theta}
\end{gather}
Adding some phantoms also fixes the vertical spacing
\begin{gather}
\begin{align}
r &= \sqrt{x^2 + y^2} \vphantom{\frac{1_{\mathstrut}}{2}}\\
f &= m a \vphantom{\frac{1_{\mathstrut}}{2}}\\
A &= \frac{\pi}{4} d^2 \vphantom{\frac{1_{\mathstrut}}{2}}
\end{align}\\
\frac{\partial P}{\partial \theta} = \frac{P_{(i+1,j)} - P_{(i-1,j)}}{2 \Delta\theta}
\end{gather}
\end{document}

If you have text between the two environments, just type it in, without blank lines in between.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
If we have some word between the two environments, it's very easy:
\begin{align}
r &= \sqrt{x^2 + y^2} \\
f &= m a \\
A &= \frac{\pi}{4} d^2
\end{align}
where
\begin{equation}
\frac{\partial P}{\partial \theta} = \frac{P_{(i+1,j)} - P_{(i-1,j)}}{2 \Delta\theta}
\end{equation}
The important thing is not leaving blank lines before and after the math display
environments.
\end{document}
A blank line can follow a display, but only if a really new paragraph begins.

amsmath/mathtools) . From reader side view math construct of number (different?) math environments is difficult to read and follow. – Zarko Aug 21 '16 at 01:36alignfor a single equation: there'sequationfor that. – egreg Aug 21 '16 at 07:40