When do total differentials cancel with partial derivatives

[sunrah]

I've just done a derivation and had to use the following

u_{b}u^{c}\partial_{c}\rho = u_{b}\frac{dx^{c}}{d\tau}\frac{\partial\rho}{\partial x^{c}} = u_{b}\frac{d\rho}{d\tau}

We've done this cancellation a lot during my GR course, but I'm not clear exactly when/why this is possible.

EDIT: is this only true in inertial coordinates?

 

[Fightfish]

Are you familiar with the multivariable chain rule
\frac{d f (x,y)}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}?
The 'cancellation' you performed there is simply a simplification using the chain rule (remember that you are using the Einstein summation convention).

 

[sunrah]

Are you familiar with the multivariable chain rule
\frac{d f (x,y)}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}?
The 'cancellation' you performed there is simply a simplification using the chain rule (remember that you are using the Einstein summation convention).

Thanks, i did notice that of course after posting

 

[HallsofIvy]

And, while it may be a useful "mnemonic", the derivative, ordinary or partial, is NOT a fraction and the "chain rule" does NOT involve "cancelling".

 

[bcrowell]

And, while it may be a useful "mnemonic", the derivative, ordinary or partial, is NOT a fraction and the "chain rule" does NOT involve "cancelling".

That's not really accurate IMO in the case of an ordinary derivative. There's some good discussion here: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio .