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I am trying to explain what implicit time dependence is and how it differs from explicit time dependence, but I'm unsure how "sound" my explanation is. Here is what I said:

Suppose I have a function $T(x,y,z,t)$ which is a temperature field of a room. If $T$ has explicit time-dependence, then even though I may stay at a fixed point, $(x,y,z)$, in the room, the value of $T$ at that point can change over time, the temperature may increase or decrease over time. Now, suppose for simplicity that the temperature at each point in the room doesn't change in time, i.e. $T$ has no explicit time dependence such that $T=T(x,y,z)$.

Then, suppose that I start walking around the room, now the temperature at each point remains fixed, but now my position $(x(t),y(t),z(t))$ is time dependent, i.e. it changes in time due to me walking around the room. As such, the temperature function $T$, that I use to measure the temperature at each point, is said to have implicit time dependence since its value changes over time, not due to the value at each given point changing over time, but because the position, $(x(t),y(t),z(t))$, at which it is evaluated, is changing over time.

Hence, for explicit time dependence, we have that $\frac{\partial T}{\partial t}\neq 0$, since even if my position remains fixed, the temperature at that point is changing over time. For implicit time dependence (and no explicit time dependence), we have that $\frac{\partial T}{\partial t}= 0$, since the value of the temperature at each point is fixed in time, however, $\frac{dT}{dt}\neq 0$, since my position is changing over time and so the temperature will implicitly change over time due to my movement around the room.

Would this be a correct explanation at all, or would something else be better?

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marked as duplicate by ACuriousMind, sammy gerbil, John Rennie, Qmechanic Jul 29 '16 at 23:55

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