Link between these velocity vector derivatives?

Imagine following a fluid particle through space. So any component of its position vector is a function of time.

x_i = x_i (t)

What is the velocity of this particle?

u_i = \frac {d x_i (t)}{dt}

(1)

However, I also have that;

u_i = u_i (x_1 (t), x_2 (t), x_3 (t), t)

(2)

Apparently equations (1) and (2) are equal, but how can this be mathematically the case? Logically it makes sense, the velocity in a specific direction is equal to the rate of change of position in that direction w.r.t time.

Similarly it logically makes sense that any velocity component is a function of all three position components and time. Its velocity varies with position, and if the fluid particle were at the same position at two different times, it may have different velocities (bringing up the time dependence).

However how can;

u_i (x_1 (t), x_2 (t), x_3 (t), t) = \frac {d x_i (t)}{dt}

Mathematically be correct? Because

x_i

is a function of time ONLY, and so where can the additional dependencies of the velocity component on

x_1, x_2, x_3

arise from when we take the derivative w.r.t time?

Reply 1

Unkempt_One

x_1, x_2

and

x_3

are also functions of time.

(edited 11 years ago)

Reply 2

Anti Elephant Mine

I know that. How can something which takes a derivative of something which is only a function of time become a function of other things too (other position coordinates)?