# Find the curvature of the curve

1. Oct 4, 2009

### undrcvrbro

1. The problem statement, all variables and given/known data
Find the curvature $$\kappa(t)$$ of the curve $$\\r(t)=(2sint)i +(2sint)j +(3cost)k$$

2. Relevant equations
$$\\\k(t)= (\left|T'(t)\right|) / (\left|r'(t)\right|)$$

3. The attempt at a solution
I found $$\\\\r'(t)= (2cost)i + (2cost)j + (-3sint)k$$

$$\\\\\ \left|r'(t)\right|= sqrt((2cost)^2 + (2cost)^2 + (-3sint)^2 \left|r'(t)\right|=sqrt((4cost)^2+(-3sint)^2) \left|r'(t)\right|=sqrt(4+-3) \left|r'(t)\right|=sqrt(1)$$

I think this is where I'm getting caught up. I wont go any further becuas I'm postive I messed up the sin cos relationship when finding the magnitude of r'(t). For all I know, I could have made another mistake along the way.

I understand the equations we are using in this course(Calc III), but I almost always find myself getting caught up on the basic mathematics.

Can any help lead me in the right direction for this problem?

2. Oct 4, 2009

### jbunniii

$$(t) = (|T'(t)|) / (|r'(t)|)$$

The curvature I am familiar with is

$$\kappa(t) = \frac{|r'(t) \times r''(t)|}{|r'(t)|^3}$$

Can you write down the definition of $T(t)$, so we can check whether our definitions are the same? Also, what does the $(t)$ on the left-hand side refer to? I guess it's a typo for $\kappa(t)$?

Last edited: Oct 4, 2009
3. Oct 4, 2009

### undrcvrbro

They are the same.

T(t)= r'(t)/ abs(r(t))

Just a different way of writing it I guess. And yeah, that's k(t).I'm sort of learning latex as I go.

4. Oct 4, 2009

### jbunniii

OK, so let's proceed from there. You made several errors in your calculation of $r'(t)$. It should be

\begin{align*}|r'(t)| &= \sqrt{(2 \cos t)^2 + (2 \cos t)^2 + (-3 \sin t)^2} \\ &= \sqrt{4 \cos^2 t + 4 \cos^2 t + 9 \sin^2 t} \\ &= \sqrt{8 \cos^2 t + 9 \sin^2 t}\end{align*}

which can't easily be simplified further. (It certainly doesn't equal $\sqrt{8 + 9}$!)

Now can you calculate $T(t)$?

P.S. In case you didn't already know, you can click on any typeset equation in these forums to see the Latex code that produced it. Very useful while learning.

5. Oct 4, 2009

### jbunniii

OK, you can make a bit more simplification than what I wrote:

\begin{align*}|r'(t)| &= \sqrt{8 \cos^2 t + 9 \sin^2 t} \\ &= \sqrt{8 \cos^2 t + 8 \sin^2 t + \sin^2 t} \\ &= \sqrt{8 (\cos^2 t + \sin^2 t) + \sin^2 t} \\ &= \sqrt{8 + \sin^2 t}\end{align*}

6. Oct 4, 2009

### undrcvrbro

So then would $T(t)$
1/(sqrt(8(sint)^2)) * (2cost, 2cost, -3sint)

So then to find T'(t) you would have to use the quotient rule, right?

Thanks for the tip. Sorry if the normal text is to hard to read. I'm getting frustrated trying to figure out how to put this into latex. It seems to only complicate things more. I'll eventually get the hang of it.