test

[imatfaal]

After Stage 4

GERRANS Simon (Orica Greenedge)(AUS) 13:41:58

 

Subaru King of the Mountain

HENAO Sergio (Team Sky)(COL)26

 

iiNet Sprint Jersey

MCCARTHY Jay (Tinkoff)(AUS)46

 

Europcar Young Rider Jersey

MCCARTHY Jay (Tinkoff)(AUS)13:42:12

 

Wilson Parking Winning Team award

CANNONDALE PRO CYCLING TEAM (USA)

 

Stage 5 McLaren Vale to Willunga Hill 151km

 

 

Looks really lumpy - but the scale is deceptive. However Willunga Hill has been decisive in the past

 

 

 

Russel van Hout and Paul Crake cross the line in 2006 after establishing a gap on Willunga Hill

[imatfaal]

[latex]t_L=\frac{1}{H_0} \int_{0}^{z}\frac{dz'}{(1+z')\sqrt {\Omega(1+z')^3+\Omega_k(1+z')^2+\Omega_{\lambda}}}[/latex]

[imatfaal]

Actually in that statement "The change in the value of a function/variable" is previously defined:

[latex]\Delta f(x)= f(x_2)-f(x_1)[/latex] or [latex]\Delta x= x_2-x_1[/latex]

But how we defined [latex]\mathrm{d} f(x)[/latex] or [latex]\mathrm{d} x[/latex] is not precisely elaborated. One may say it's the same as a change in the variable/function when the change is small enough but what "small enough" means here again isn't precisely elaborated. I actually attempted to think of a definition for differential as like it's simply the limit of a difference as the difference approaches zero:

[latex]\mathrm{d}x= \lim_{\Delta x \to 0}\Delta x[/latex]

But because of the so called "Archimedean property" of the "Real number system", that makes a differential simply zero:

[latex]\mathrm{d}x= 0[/latex]

and it's not logical.

 

Can we make this situation better?

 

[studiot]

For those who might be wondering what this is all about,

 

Posting a reply went badly wrong for me this afternoon in this thread.

 

http://www.scienceforums.net/topic/102139-rigorous-definition-of-differential/

 

and imatfaal is very kindly helping out.

 

Please note there is still something wrong there.

Edited January 5, 2017 by studiot

[StringJunky]

Just a note: If your cursor gets stuck in the quote box, double-tap Enter with the cursor at the end of the quote. It should pop out onto the next line outside the box. If not, then try clicking outside the box with your mouse or touchpad. Either one of the two works for me depending what mood the software is in.

Edited January 5, 2017 by StringJunky