Confusion with gradient and divergence of vector fields and scalar fields

Question

In a proof, my teacher used the following assumption for an approximation: $$ |\nabla n(\boldsymbol{r}, t)| \ll\left|\nabla \partial_{t} \boldsymbol{\tilde{r}}(\boldsymbol{r}, t)\right| $$ where $n$ denotes electron density and $\tilde{\boldsymbol{r}}=\boldsymbol{r}_{0} e^{i \boldsymbol{k} r-i \omega t}$ is the temporal derivative of electron position. However, since $n$ is a scalar quantity (for each $\mathbf{r},t$) it is a scalar field. The gradient of a scalar field is undefined. I then thought he might mean the divergence. But the divergence of a scalar field is a vector field, while the gradient of a vector field (right-hand side of the equation) is a tensor quantity. How can the equation above ever make sense when $n$ is a scalar field and $\boldsymbol{\tilde{r}}$ a vector field?