How to reduce space between two math environments?

Consecutive math displays should be avoided. If you really must group aligned display math and non-aligned content, use a single display math environment (here, gather*) and use an internal math environment to perform the alignment (here, aligned):

\documentclass[a4paper,14pt]{extreport}
\usepackage{amsmath}
\begin{document}
\begin{gather*}
\begin{aligned}
     r &= \sqrt{x^2 + y^2} \\
     f &= m a \\
     A &= \frac{\pi}{4} d^2
\end{aligned} \\
     \frac{\partial P}{\partial \theta} = \frac{P_{(i+1,j)} - P_{(i-1,j)}}{2 \Delta\theta}
\end{gather*}
\end{document}

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.