Multivariable class, we'll be starting curvature

Main Question or Discussion Point

Next week in my multivariable class, we'll be starting curvature, and, nerd that I am, I looked ahead to learn it ahead of time. I can usually at least understand the basics of a new concpet by myself, but curvature really threw me off. Maybe my brain's not right for it, maybe the book sucks, but I know my teacher sucks, so I'm pretty much going to have to learn it myself.

I know that curvature is $\kappa=|d\phi/ds|$, where $\phi$ is the angle between the curve's tangent vector and the horizontal, and s is the arc length.

I also get that the way to do this is to make it $\kappa=|d\phi/dt/(ds/dt)|$, I get lost, however, in actually finding a good equation for $\phi$.

Does anyone have any tips, resources, or advice?

From what I've learned, that equation is hard to work with. Have you seen it as this?:

$$\kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^{3}}$$

It's much easier to work with (r is the position vector). To show that the two definitions are equal, use the following fact:

$$\mathbf{r'}=\frac{ds}{dt}\mathbf{T}$$

That's a lot better :D . Thanks! That actually makes sense in a twisted sort of way, and it's a lot easier to use.