PX275 - E2d - scalar fields

the divergence theorem can also be used with scalar fields using a maths trick
consider a scalar field, $ϕ$ , so a vector can be defined as: $\vec{F} = ϕ \vec{c} = ϕ (c_{x} \hat{i} + c_{y} \hat{j} + c_{z} \hat{k})$
the divergence of this vector:

\begin{aligned} \vec{\nabla} \cdot \vec{F} & = \frac{\partial}{\partial x} (ϕ c_{x}) + \frac{\partial}{\partial y} (ϕ c_{y}) + \frac{\partial}{\partial z} (ϕ c_{z}) \\ = \frac{\partial ϕ}{\partial x} c_{x} + ϕ_{x} \frac{\partial c_{x}}{\partial x} + \dots \end{aligned}

∴ \vec{\nabla} \cdot \vec{F} = \vec{\nabla} \cdot (ϕ \vec{c}) = \vec{c} \cdot \vec{\nabla} ϕ + ϕ (\vec{\nabla} \cdot \vec{c})

\vec{\nabla} \cdot \vec{F} = \vec{c} \cdot \vec{\nabla} ϕ

\begin{aligned} \iint_{S} ϕ \vec{c} \cdot d \vec{s} & = ∭_{V} (\vec{c} \cdot \vec{\nabla} ϕ) d V \\ \vec{c} \iint ϕ d \vec{s} & = \vec{c} ∭_{V} \vec{\nabla} ϕ d V \end{aligned}

∴ \iint ϕ d \vec{s} = ∭_{V} \vec{\nabla} ϕ d V