Originally Posted by

**HallsofIvy** If it is given that T is a function of x, y, and z only, while x, y, and z are functions of t, then

$\displaystyle \frac{dT}{dt}= \frac{\partial T}{\partial x}\frac{dx}{dt}+ \frac{\partial T}{\partial y}\frac{dy}{dt}+ \frac{\partial T}{\partial z}\frac{dz}{dt}$

If T is a function of x, y, z, **and** t, then

$\displaystyle \frac{dT}{dt}= \frac{\partial T}{\partial t}+ \frac{\partial T}{\partial x}\frac{dx}{dt}+ \frac{\partial T}{\partial y}\frac{dy}{dt}+ \frac{\partial T}{\partial z}\frac{dz}{dt}$