Does this derivative have a name?

  • Thread starter pellman
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  • #1
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Given a function F(x,t) where x is a function of t, we write the total derivative as

[tex]\frac{dF}{dt}=\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial t}[/tex]

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?

[tex]\frac{\partial F}{\partial x}\frac{dx}{ds}+\frac{\partial F}{\partial s}[/tex]
[tex]\frac{\partial F}{\partial x}\frac{dx}{dt}+\frac{\partial F}{\partial t}[/tex]
 

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  • #2
Office_Shredder
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The total derivative is really just the chain rule in multiple variables. With this in mind your two quantities could be described as [itex] \frac{d F}{ds}[/itex] and [itex] \frac{dF}{dt} [/itex] and nobody would flinch
 
  • #3
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I like that answer. Thanks!
 
  • #4
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The total derivative is really just the chain rule in multiple variables. With this in mind your two quantities could be described as [itex] \frac{d F}{ds}[/itex] and [itex] \frac{dF}{dt} [/itex] 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 [itex]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[/itex]. So you can see that [itex]\frac{dF}{ds}[/itex] and [itex]\frac{dF}{dt}[/itex] 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 [itex]dF=dG=\frac{\partial G}{\partial s}ds+\frac{\partial G}{\partial t}dt[/itex]. So we can call the two expressions [itex]\frac{\partial G}{\partial s}[/itex] and [itex]\frac{\partial G}{\partial t}[/itex].
 
  • #5
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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.
 
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