Arc length formal proofs for 2016

[zsteve]

Hi guys,
I had too much time so I decided to play around with and write up formal proofs for arc length formulas which are relevant to the new Specialist 3/4 study design. (Still amazed by how pretty the typesetting is, but yeah )
I've included a proof of arc length for a scalar valued function of a single variable (which I think receives rigorous treatment and proof), but also of the general case for a parametric equation. Here I construct the proof using a vector-valued function of a single real parameter in , and then give special cases for and , expressed in vector form (instead of using magnitude of the vector function). This appears in the textbook without rigorous proof (rather, an 'intuitive' fudging). So I hope my method sheds some light on exactly how and why it works.

Enjoy, and remember - this was the product of too much free time after exams

Mods: if appropriate please move to correct subtopic/category

Edit: updated link https://onedrive.live.com/redir?resid=2692D5EC8060E581!6983&authkey=!AOWnn7xS-PD6CBo&ithint=file%2cpdf

[lzxnl]

Hi guys,
I had too much time so I decided to play around with and write up formal proofs for arc length formulas which are relevant to the new Specialist 3/4 study design. (Still amazed by how pretty the typesetting is, but yeah )
I've included a proof of arc length for a scalar valued function of a single variable (which I think receives rigorous treatment and proof), but also of the general case for a parametric equation. Here I construct the proof using a vector-valued function of a single real parameter in , and then give special cases for and , expressed in vector form (instead of using magnitude of the vector function). This appears in the textbook without rigorous proof (rather, an 'intuitive' fudging). So I hope my method sheds some light on exactly how and why it works.

Enjoy, and remember - this was the product of too much free time after exams

Mods: if appropriate please move to correct subtopic/category

The idea is right. And the effort is certainly commendable.
Only thing I'd add is that your infinite sum is the Riemann sum definition of an integral. You really need to specify that your sum defines an integral; otherwise, I'm left going, 'what?'

[Sine]

Attachment is not found, could you re-upload it?

[zsteve]

Bump
I realise that most of the SM class of 2016 has covered arclength, this might be of interest again!