Substantial Derivative

Reference to Fluid Simulation for Computer Graphics.
In fluid simulation, we have a velocity vector field \mathbf{u}(\mathbf{x}, t)=[u(\mathbf{x}, t), v(\mathbf{x}, t), w(\mathbf{x}, t)] which depends on position \mathbf{x}=[x, y, z] and time t. We can calculate its full derivative for time t base on the chain rule:
\begin{aligned}  \frac{d}{dt}\mathbf{u}(\mathbf{x},t)   &= \frac{\partial{\mathbf{u}}}{\partial{t}}\frac{dt}{dt} + \nabla\mathbf{u}\cdot\frac{d\mathbf{x}}{dt}\\  &= \frac{\partial{\mathbf{u}}}{\partial{t}} + \frac{\partial{\mathbf{u}}}{\partial{x}}\frac{dx}{dt} + \frac{\partial{\mathbf{u}}}{\partial{y}}\frac{dy}{dt} + \frac{\partial{\mathbf{u}}}{\partial{z}}\frac{dz}{dt} \\  &= \frac{\partial{\mathbf{u}}}{\partial{t}} + u\frac{\partial{\mathbf{u}}}{\partial{x}} + v\frac{\partial{\mathbf{u}}}{\partial{y}} + w\frac{\partial{\mathbf{u}}}{\partial{z}} \\  &= \frac{\partial{\mathbf{u}}}{\partial{t}} + \mathbf{u}\cdot\nabla\mathbf{u} \\  &= \frac{D\mathbf{u}}{Dt}    \end{aligned}

Which is the substantial derivative (or material derivative).

If you want to fully expand this derivative:

\begin{aligned}  \frac{D\mathbf{u}}{Dt}  &= \frac{\partial{\mathbf{u}}}{\partial{t}} + \mathbf{u}\cdot\nabla\mathbf{u} \\    &= \begin{bmatrix}  \partial{u}/\partial{t} + \mathbf{u}\cdot\nabla{u}\\  \partial{v}/\partial{t} + \mathbf{u}\cdot\nabla{v}\\  \partial{w}/\partial{t} + \mathbf{u}\cdot\nabla{w}  \end{bmatrix}\\  &= \begin{bmatrix}  \frac{\partial{u}}{\partial{t}}+u\frac{\partial{u}}{\partial{x}} + v\frac{\partial{u}}{\partial{y}} + w\frac{\partial{u}}{\partial{z}}\\  \frac{\partial{v}}{\partial{t}}+u\frac{\partial{v}}{\partial{x}} + v\frac{\partial{v}}{\partial{y}} + w\frac{\partial{v}}{\partial{z}}\\  \frac{\partial{w}}{\partial{t}}+u\frac{\partial{w}}{\partial{x}} + v\frac{\partial{w}}{\partial{y}} + w\frac{\partial{w}}{\partial{z}}  \end{bmatrix}    \end{aligned}

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