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.