In this chapter, we only considered path integrals for one-dimensional systems, but in some cases it is straightforward to calculate the path integral for a system in three spatial dimensions. As an example of this, calculate the path integral for the three-dimensional harmonic oscillator, which has potential energy

\[\begin{equation*}U(\vec{r})=\frac{m \omega^{2}}{2} \vec{r} \cdot \vec{r}=\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}+z^{2}\right) \tag{11.161}\end{equation*}\]

Don't forget that both the position and momentum of the particle in the potential have \(x, y\), and \(z\) components.