We derive from first principles analytic relations for the second- and third-order moments of m, the spatial gradient of fluid velocity u, m = del u, in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.