13.3: Arc Length & Curvature (1/2)

Objectives:
10. Find the length of a space curve.
11. Parameterize a curve with respect to arc length.
12. Give the properties of arc length parameterizations.
13. Define and compute curvature for curves in 2-space and 3-space.

[@1light4love]

my College Calc3 with Engineering Applications Professor just spent 2 & 1/2 hours attempting to explain this exact section and I guarantee you that not one person had the slightest understanding of what he was talking about. (which is normal, unfortunately.) thank GOD for you, Professor.... and thank you for your straightforward, easily understood approach. 🙏 (Our first Exam is next week!)

[@alexandraniedden5337]

Good luck!

[@xiaoyangwang9191]

I'm not saying my cal3 professor is bad at teaching, however, you have explained it clearly that I'm actually understanding it, thanks so much for these cal3 videos, it really helped me.

[@butx1555]

29:00 my professor used a slightly different method. He did not find the arc length parametrization, instead, he used K(t)= the norm of T'(t)/the norm of r'(t) Update: 33:30 you explained my teacher's method. You are literally the best teacher!

[@charliezhou6514]

be my professor

[@sundaymanali5854]

great lectures.

[@brendanbuckner2561]

At 29:28 why did the "a" values disappear when taking the derivative of vector r(s)?

[@alexandraniedden5337]

When taking the derivative of the first term, for example, remember that a is a constant. The derivative of acos(s/a) would be a*-sin(s/a)*1/a (1/a from Chain Rule). That a*1/a gives us 1.

[@nolfre1382]

I love your videos but this one in specific was way too fast for me to understand everything. Maybe it's just me but i got frustrated for the first time. Anyways thank you, this actually helps.

[@butx1555]

Do you go to school in Madison?

[@nolfre1382]

@@butx1555 nop

[@devanarayanababu1996]

ms.budden why is there a need to parameterise the arc length? (what is the use?)

[@alexandraniedden5337]

We often choose to parameterize in terms of arc length for analysis purposes. Using s as a variable is significantly more meaningful (and convenient) as compared to an arbitrary variable, like u. Around 14:00, the example at the bottom shows one way that arc parametrization is helpful. If I move 5 units, where am I now?

[@backup0197]

@@alexandraniedden5337 oohk......thank you ma'am

[@nuhaalnahwi5618]

For example 2 why is the integral from 0 to t? how did u pick which point from the reference point?

[@alexandraniedden5337]

We are asked for the arc length parametrization, which is a general vector rather than a specific vector. This is why the upper limit is t, rather than a number (also follows the arc length parametrization formula). To find the lower bound, we must consider the reference point. Setting (1, 0, 0) = (cost, sint, t) will help us find the t value at which this point occurs. Hence cost = 1, sint = 0 and t = 0. The only place this happens is t = 0.

[@uzairahmed.1790]

Can you please share the notes of chapter 12 and 13?

[@alexandraniedden5337]

Chapter 12: kzbin.info/aero/PLGOk2-zeLtjBuFyoBkgdoRh0qhuaqrBdi Chapter 13: kzbin.info/aero/PLGOk2-zeLtjAG7o5KUBbGAgaKyAN4ZFd6 A few lectures may be missing, but this should be most of them. Good luck!

[@uzairahmed.1790]

@@alexandraniedden5337 Thanks alot! These lectures are far better than my college lectures.

[@jakeb110]

How did you get 5pi in ex1 I've been working on it and I can't seem to get it.

[@alexandraniedden5337]

Under the square root, 9sin^2t + 9cos^2t simplifies to 9 (remember that sin^2t + cos^2t = 1). Then we have the integral of sqrt(9 + 16) which gives us the integral of 5 from 0 to pi. I hope that makes sense. Good luck!

[@jakeb110]

Also for the last problem formula 3 I keep getting 2(t^2+ 2t+ 2) instead of 2(t^2+2) I'm getting lost on how you get rid of the 16t^2 inside the root im pretty sure.

[@alexandraniedden5337]

@@jakeb110 The 4t^4 + 16t^2 + 16 factors. I factored out a 4 leaving (t^4 + 4t^2 + 4) which then factors to 4(t^2 + 2)^2.

[@jakeb110]

@@alexandraniedden5337 Those are more clear now, thank you very much!

[@MdimamMdimam-sg9nq]

aslamualikum mam, where are from mam?

[@fourn7wenty]

That last question though 🥲

[@butx1555]

41:31 yaas I got that😍

[@nishnababy]

Wtf....

[@n.a3642]

What?

[@sundaymanali5854]

wut