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Let $u:\mathbb R^3\xrightarrow{}\mathbb R$. Verify that
$$\nabla \cdot \nabla u = u_{xx}+u_{yy}+u_{zz}.$$
Let $u:\mathbb R^3\xrightarrow{}\mathbb R$. Then
$$\nabla\cdot\nabla u = \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle \cdot \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle u$$
$$ = \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle \cdot \left\langle \frac{\partial u}{\partial x},\frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\right\rangle$$
$$= \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = u_{xx} + u_{yy} + u_{zz}.$$
Comments
Let $u:\mathbb R^3\xrightarrow{}\mathbb R$. Then
$$\nabla\cdot\nabla u = \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle \cdot \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle u$$
$$ = \left\langle \frac{\partial}{\partial x},\frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle \cdot \left\langle \frac{\partial u}{\partial x},\frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\right\rangle$$
$$= \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = u_{xx} + u_{yy} + u_{zz}.$$