# Curvature of polar curve

1. Sep 27, 2015

### WK95

1. The problem statement, all variables and given/known data
Given the polar curve r=e^(a*theta), a>0, find the curvature K and determine the limit of K as (a) theta approaches infinity and (b) as a approaches infinity.

2. Relevant equations
x=r*cos(theta)
y=r*sin(theta)
K=|x'y''-y'x''|/[(x')^2 + (y')^2]^(3/2)

3. The attempt at a solution
I've tried converting the polar curve using the first equation, solving for their first and second derivatives, then plugging them into equation 3 but that gets very, very long.

So next, I apply the properties of limits to the relevant equations. However, I get stuck when I need to find the limit of asin(x), acos(x), sin(x), cos(x) as x approaches infinity. What now?

2. Sep 27, 2015

### Staff: Mentor

You know sin(x) and cos(x) are limited. That should be sufficient.

If asin is the inverse sine, then asin(x) with x to infinity shouldn't occur. If it is a*sin(x), then see above.

3. Sep 27, 2015

### WK95

I'm getting 0 for the limit of x', x'', y', y'' as a approaches infinity and when theta approaches infinity.

4. Sep 27, 2015

### Staff: Mentor

A curvature that approaches zero makes sense.

5. Sep 27, 2015

### WK95

Yes, but then I'd end up with a 0 in the denominator in the third equation which wouldn't work. However, that would require me to find a work around and solve for the limit some other way so I don't get 0/0. I haven't been able to figure out what I have to do to get around that.

I am expecting the curvature to be 0 in both cases when a or theta approaches infinity so I'm expecting the numerator to be 0.

6. Sep 27, 2015

### SteamKing

Staff Emeritus
Instead of using the radius of curvature formula for cartesian coordinates, why don't you try the equivalent formula for polar coordinates?