unit tangent vector T and curvature

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[runic cape]

do the same thing I said for the last time you asked this

[runic cape]

t not x, and assuming sec(t) > 0

so $T=\left(\frac{1}{\abs{\sec(t)}},\frac{-\tan(t)}{\abs{\sec(t)}}\right)$

[silk coral]

Omegabet_

[runic cape]

derivative of cos(t) isnt sin(t)

(again, assuming sec(t)>0)

[wicked dome]

Oh it's - sin t right?

[runic cape]

as your calc1 class would tell you, yes

[wicked dome]

So must put modulus?

[runic cape]

well you havent specified the domain of the curve

actually sorry, you require cos(t) > 0, so yes, it's just sec(t)

but anyway you get $T=(\cos(t),-\sin(t))$

[silk coral]

Omegabet_

[wicked dome]

How do you get -sin t though? From -tan^2 (t) / sec (t) + sec (t)

[runic cape]

that's T

not T'

-tan(t)/sec(t) = -sin(t)/cos(t) * cos(t) = -sin(t)

[wicked dome]

Oh

[runic cape]

so it's easy to see T'=(-sin(t),-cos(t)), then the formula for curvature computes curvature

[runic cape]

You have $\kappa=\frac{\norm{r'\times r''}}{\norm{r'}^3}$

[wicked dome]

Oh

[silk coral]

Omegabet_

[wicked dome]

But since we already have unit tangent vector T, so we can just use |T'(t) | / | r'(t) |?

From my understanding

[runic cape]

yeah you can use that formula too

[runic cape]

I just know for arbitrary regular curves kappa has that formula

but yes, by Frenet, $T'=\kappa v N\to \norm{T'}=\kappa v\norm{N}=\kappa v$, so $\kappa=\frac{\norm{T'}}{\norm{T}}$

[silk coral]

Omegabet_

[runic cape]

since N is unit

[wicked dome]

Just curious if I use the other formula

[runic cape]

x is cross product since these are vectors

not component wise mutliplication

[wicked dome]

What does it mean

[runic cape]

... cross product

if you dont know cross product then just use your formula

[wicked dome]

I kinda know it, but I don't know how to use it if I got only two component, usually it has 3 ( x, y, z) right?

[runic cape]

you embed it into the plane z=0

but anyway just use |T'|/|T|

[wicked dome]

Oh okayh

[wicked dome]

The answer for curvature, is it correct?

[runic cape]

looks correct

[wicked dome]

Oh alright then thanks for being patient with me btw, I struggle with calculus actually, so I appreciate your kindness 😊