Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Does this derivative have a name?

  1. May 11, 2012 #1
    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]
     
  2. jcsd
  3. May 11, 2012 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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
     
  4. May 13, 2012 #3
    I like that answer. Thanks!
     
  5. May 13, 2012 #4
    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].
     
  6. May 13, 2012 #5
    Thanks. This is precisely how I usually deal with it my own notes when I want to be careful.
     
    Last edited: May 14, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook