Given the formula for the gradient in spherical coordinates $\nabla=\frac{\partial}{\partial r}ê_r+\frac{1}{r}\frac{\partial}{\partial \varphi}ê_{\varphi}+\frac{1}{r \sin{\varphi}}\frac{\partial}{\partial \vartheta}ê_{\vartheta}$ and a vector field in spherical coordinates $\vec{v}(\vec{r})=v_rê_r+v_{\varphi}ê_{\varphi}+v_{\vartheta}ê_{\varphi}$ I am trying to show the formula for divergence given by$$\textrm{div }\vec{v}= \frac{1}{r^2}\frac{\partial (r^2 v_r)}{\partial r}+\frac{1}{r \sin \vartheta}(\frac{\partial(v_{\vartheta}\sin \vartheta)}{\partial \vartheta}+\frac{\partial v_{\varphi}}{\partial \varphi})$$ while using the relation $\textrm{div } \vec{v} = \nabla \cdot \vec{v}$.
Since in non-cartesian coordinates I can't use the regular dot product, my approach was to evaluate $(\frac{\partial}{\partial r}ê_r+\frac{1}{r}\frac{\partial}{\partial \varphi}ê_{\varphi}+\frac{1}{r \sin{\varphi}}\frac{\partial}{\partial \vartheta}ê_{\vartheta}) \cdot (v_rê_r+v_{\varphi}ê_{\varphi}+v_{\vartheta}ê_{\varphi})$ where I could make use of the fact that the unit vectors are equal to one when squared and equal to zero when multiplied with a different unit-vector. However, since that only leaves me with $\frac{\partial v_r}{\partial r}+\frac{1}{r}\frac{\partial v_{\varphi}}{\partial \varphi}+\frac{1}{r \sin \vartheta}\frac{\partial v_{\vartheta}}{\partial \vartheta}$, I'm really wondering what I am missing? Am I doing the multiplication wrong?