Differentiation help...

Phymath

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)

dextercioby

U messed up the tex code...

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...

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

Daniel.

Phymath

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?

dextercioby

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.

Da

Phymath

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...

dextercioby

Thermodynamics:
[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]

Daniel.

Galileo

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]
or
[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.

Phymath

i see good explaination! thanks