Does this derivative have a name?

[pellman]

Given a function F(x,t) where x is a function of t, we write the total derivative as

\frac{dF}{dt}=\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial t}

Now what if we have two parameters, F(x,s,t) where x is a function of both s and t. What do we call the following quantities and is there a conventional notation for them?

\frac{\partial F}{\partial x}\frac{dx}{ds}+\frac{\partial F}{\partial s}
\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial t}

 

[Office_Shredder]

The total derivative is really just the chain rule in multiple variables. With this in mind your two quantities could be described as \frac{d F}{ds} and \frac{dF}{dt} and nobody would flinch

 

[pellman]

I like that answer. Thanks!

 

[lugita15]

The total derivative is really just the chain rule in multiple variables. With this in mind your two quantities could be described as \frac{d F}{ds} and \frac{dF}{dt} and nobody would flinch

Well, consider me a flincher. Let me lend some perspective from the traditional notions of differentials, which I think is often a useful way to think about things.

Applying the chain rule, the total differential of F is given by dF=\frac{\partial F}{\partial s}ds+\frac{\partial F}{\partial t}dt+\frac{\partial F}{\partial x}dx=\frac{\partial F}{\partial s}ds+\frac{\partial F}{\partial t}dt+\frac{\partial F}{\partial x}\left(\frac{\partial x}{\partial s}ds+\frac{\partial x}{\partial t}dt\right)=\left(\frac{\partial F}{\partial s}+\frac{\partial F}{\partial x}\frac{\partial x}{\partial s}\right)ds+\left(\frac{\partial F}{\partial t}+\frac{\partial F}{\partial x}\frac{\partial x}{\partial t}\right)dt. So you can see that \frac{dF}{ds} and \frac{dF}{dt} would not be the right name for these two parenthetical expressions. So what would you call them? To avoid abuse of notation, let me define a function G(s,t)=F(s,t,x(s,t)). In other words, we're just not including x as a variable anymore. In that case, we have dF=dG=\frac{\partial G}{\partial s}ds+\frac{\partial G}{\partial t}dt. So we can call the two expressions \frac{\partial G}{\partial s} and \frac{\partial G}{\partial t}.

 

[pellman]

define a function G(s,t)=F(s,t,x(s,t)).

Thanks. This is precisely how I usually deal with it my own notes when I want to be careful.