Finding Curvature of Vector Function

[Bashyboy]

Homework Statement

Find the curvature K of the curve, where s is the arc length parameter:

\vec{r}(t) = \langle 2 \cos t , 2 \sin t, t \rangle

Homework Equations

s(t) = \int_a ^t ||\vec{r}'(u)||du

The Attempt at a Solution

I know I need to find the arc length function, in order to find the curvature function; however, I am unsure as to what I should choose a to be for the lower limit of the integral.

 

[pasmith]

You need to use the Frenet-Serret formulae and the chain rule.

You should not need to perform any integrations.

 

[ehild]

Homework Statement

Find the curvature K of the curve, where s is the arc length parameter:

\vec{r}(t) = \langle 2 \cos t , 2 \sin t, t \rangle

Homework Equations

s(t) = \int_a ^t ||\vec{r}'(u)||du

The Attempt at a Solution

I know I need to find the arc length function, in order to find the curvature function; however, I am unsure as to what I should choose a to be for the lower limit of the integral.

Choose it so that s(0)=0.

 

[Bashyboy]

So, the choice is arbitrary? If so, why?

 

[ehild]

So, the choice is arbitrary? If so, why?

Why not? Just as you can start time at any instant, you can measure an arc length from any point on a line. Usually the arc length is connected to the trajectory of an object if time is involved, that is why I suggested s(0)=0. But you also can keep "a" in the expression s(t). Solve the problem and you will see that"a" cancels.

ehild