Gradient Proof

Given a vector function, I show that \vec{F}(x,y,z,t) , I show that d\vec{F} = (d\vec{r} \cdot \nabla) \vec{F} + \frac{\partial \vec{F}}{\partial t}  by writing d\vec{F} in terms of independent variations in the x, y, z and t direction. I write d\vec{F} as a sum of four increments, one purely in the x direction, the y direction, the z direction and the t direction as follows:

$latex dF(x,y,z,t)

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