Is dv/dt = ∂v/∂r + ∂v/∂t correct when velocity depends on position and time?

[sniffer]

i just want simple explanation of this.
velocity v is a function of r and t.
then my professor, in order to derive relationship of something, he wrote:
$$\frac{dv}{dt}=\frac{\partial v}{\partial r}+\frac{\partial v}{\partial t}$$
then assuming small spatial variation,
$$\frac{dv}{dt} \approx \frac{\partial v}{\partial t}$$
my question: is the first equation above OK?
is it standard calculus? i am weak in this. please help.


thanks.

 

[Mulder]

look up the chain rule; when v = v(r(t),t),


$$\frac{dv}{dt}=\frac{\partial v}{\partial r}\frac{dr}{dt}+\frac{\partial v}{\partial t}$$

then you can see the following argument makes more sense?

 

[sniffer]

thanks. i just want to clarify again, does it mean dr/dt is small, i.e. small velocity?