  We derive simple analytical expressions for the particle density $\rho(r)$
and the kinetic energy density $\tau(r)$ for a system of noninteracting
fermions in a $d-$dimensional isotropic harmonic oscillator potential. We test
the Thomas-Fermi (TF, or local-density) approximation for the functional
relation $\tau[\rho]$ using the exact $\rho(r)$ and show that it locally
reproduces the exact kinetic energy density $\tau(r)$, {\it including the shell
oscillations,} surprisingly well everywhere except near the classical turning
point. For the special case of two dimensions (2D), we obtain the unexpected
analytical result that the integral of $\tau_{TF}[\rho(r)]$ yields the {\it
exact} total kinetic energy.
