Divergence applied to vector field, question

Question

Divergence is defined as a scalar valued function: $$ \left. \operatorname{div} \mathbf{F} \right|_\mathbf{x_0} = \lim_{V \to 0} \frac{1}{|V|} \int\int_{\scriptstyle S(V)} \mathbf{F} \cdot \mathbf{\hat n} \, dS $$ where $\mathbf F$ is a vector field and $V$ an infinitesimal volume.

also defined:

$$ \nabla \cdot \mathbf F $$

I'm confused by the second definition and how it produces a scalar result. The other definition is: $$ \mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k} $$ $$ \nabla\cdot\mathbf{F} = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}. $$

The partial derivatives $\frac{\partial F_q}{\partial y}$ are of a vector valued function $\mathbf F$. Does anyone know how this results in a scalar result?

Answer 1

The partial derivatives $\dfrac{\partial F}{\partial x_i}$ are the partial derivatives of the components of your vector field. Your vector field is $\displaystyle \mathbf{\vec F}= \sum F_i \mathbf{\hat e}_i$.