How to show that the Jeans Criterion for Mass, Radius and Density are equivalent?
What you want to do is use the relation between Mass, Radius, and Density. The proper expression should be

$$\rho_c = \frac{M_c}{(4/3) \pi R_c^3}$$

Your three Jeans conditions are $M_c &gt; M_J$, $R_c &gt; R_J$, and $\rho_c &gt; \rho_j$. By using the relation above, you can transform any of the three conditions into one of the others.

For example, going from $M_c &gt; M_J \rightarrow R_c &gt; R_J$ is done as follows.

$$M_c = \frac{4}{3}\pi R_c^3 \rho_c &gt; (\frac{5kT}{G \mu m_H})^{3/2} (\frac{3}{4 \pi \rho_c}) ^{1/2}$$

Now, solving for $R_c$ gives

$$R_c^3 &gt; (\frac{5kT}{G \mu m_H})^{3/2} (\frac{3}{4 \pi \rho_c}) ^{3/2}$$

$$R_c &gt; (\frac{15kT}{4 \pi G \mu m_H \rho_c})^{1/2} \equiv R_J$$

The other transformations follow in a similar manner.