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A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is $\vec{V} = \left ( 2x\right ) \hat{i} + \left ( y + 3t \right )\hat{j}$, and the temperature field in the fluid is $T = 2x^{2} + xy + 4t$, where $x$ and $y$ are the spatial coordinates, and $t$ is the time. The time rate of change of temperature recorded by the probe at $\left ( x = 1, y = 1, t = 1 \right )$ is _____________.

  1. $4$
  2. $0$
  3. $18$
  4. $14$
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