How Does Differentiation Relate to Limits and Approximations?

[Phymath]

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

what does $$\Delta x_k$$ really repersent? graphiclly and with an example.

why is

$$df = \sum \frac{\partial{f}}{\partial{x_i}} dx_i$$
using $$dx_i$$ 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 $$\Delta x_k$$ how do u calculate that?

 

[dextercioby]

I don't know what the Taylor expanision of a function of multiple variables means, can u elaborate? Still what is $$\Delta x_k$$ how do u calculate that?

Then i'll have to advise you to read a calculus book on multiple variable calculus,where I'm sure you're going to 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:
$$dS= \frac{dU}{T}+pdV+\mu dN+...$$

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

Daniel.

 

[Galileo]

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

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

$$\frac{d}{dx}f(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=f'(x)$$

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

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

The h in this case is what the $\Delta x_k$ represents in your multivariable case.

 

[Phymath]

i see good explanation! thanks