1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Differentiation help

  1. Feb 7, 2005 #1
    I don't understand how
    [tex]\frac{\partial{f}}{\partial{x_k}}(\vec{a}) = \lim_{\epsilon\rightarrow 0} \frac{f(a_1,...,a_k+\epsilon,...,a_n)-f(a_1,...,a_k,...,a_n)}{\epsilon}[/tex]
    can be equal to...
    [tex]\Delta_k f = f(a_1,...,a_k + \epsilon,...,a_n)-f(a_1,...,a_k,...,a_n) \approx \Delta x_k \frac{\partial{f}}{\partial{x_k}}[/tex]
    and why is it approximately?

    what does [tex] \Delta x_k [/tex] really repersent? graphiclly and with an example.

    why is

    [tex] df = \sum \frac{\partial{f}}{\partial{x_i}} dx_i [/tex]
    using [tex] dx_i [/tex] what does that graphicly represent? and what is Legendre transformation and what is its significance. Please try to explain (calc I-III, Linear algebra background)
    Last edited: Feb 7, 2005
  2. jcsd
  3. Feb 7, 2005 #2


    User Avatar
    Science Advisor
    Homework Helper

    U messed up the tex code... :rolleyes:

    At the first,it's simply the partial derivative's definition,though improperly written,while at the second,it's simply the first term (linear in differentials) of the Taylor expansion of a function of multiple variables.

    Graphically,it's difficult to describe in words.Imagine a surface.The partial derivative wrt "x" at a point on the surface is nothing else but the tangent of the angle made by a tangent line in that point to the curve delimited by the Oxz plane and the surface and the Ox axis...Really disgusting... :yuck:

    Legendre transformations are fundamental in physics,yet a description of them cannot be given within a couple of lines.

  4. Feb 7, 2005 #3
    I don't know what the Taylor expanision of a function of multiple variables means, can u elaborate? Still what is [tex]\Delta x_k [/tex] how do u calculate that?
  5. Feb 7, 2005 #4


    User Avatar
    Science Advisor
    Homework Helper

    Then i'll have to advise you to read a calculus book on multiple variable calculus,where i'm sure you're gonna find the Taylor series explained much better than i'd succed if i were to try to.
    Those delta's are small variations,you do not calculate them by any mean,they're an analogus for the differentials.

  6. Feb 7, 2005 #5
    i see so is there an example u can give me that uses the total differential or where I'd need to? Such as in phyiscs...
  7. Feb 7, 2005 #6


    User Avatar
    Science Advisor
    Homework Helper

    [tex] dS= \frac{dU}{T}+pdV+\mu dN+... [/tex]

    [tex] dS=(\frac{\partial S}{\partial U})_{V,N,...} dU +(\frac{\partial S}{\partial V})_{U,N,...} dV+(\frac{\partial S}{\partial N})_{U,V,...} dN+... [/tex]

  8. Feb 8, 2005 #7


    User Avatar
    Science Advisor
    Homework Helper

    I`ll take the singe variable case, because the reasoning is similar.

    [tex]\frac{d}{dx}f(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=f'(x)[/tex]

    which means, that for any [itex]\epsilon>0[/itex] we can find a [itex]\delta[/itex], such that:
    [tex]|h|<\delta \Rightarrow \left|\frac{f(x+h)-f(x)}{h}-f'(x)\right|<\epsilon[/tex].

    So if we make h small enough, it will be close enough to f'(x). Therefore, for small h:
    [tex]\frac{f(x+h)-f(x)}{h} \approx f'(x)[/tex]
    [tex]f(x+h)-f(x) \approx hf'(x)[/tex]

    The h in this case is what the [itex]\Delta x_k[/itex] represents in your multivariable case.
  9. Feb 8, 2005 #8
    i see good explaination! thanks
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook