dT/ds - rate of change of direction. ds/dt - rate of change of distance traveled along the path

With all of these curvature videos, I can't remember a time except for once when I needed curvature as an engineer. I had some airfoil coordinates that were noisy, and that noise ends up as ripples in pressure coefficient in invisvid aerodynamics codes. I tried a variety of smoothing methods. But perhaps the most interesting one was to compute the curvature numerically from the coordinates, pass the raw curvature through a low-pass digital filter, and then numerically integrate the smoothed curvature to get a smoothed T. From a smoothed T, you can calculate smoothed (x,y) coordinates again. This was something like 30 years ago, so I don't remember all the details. But the problem i think I ran into was getting the trailing edge coordinates to be coincident. The first trailing edge coordinate has no error because it's before the first integration step. But the other trailing edge coordinate is the last integration step with all of the numerical integration error. And since the curvature profiles are different, even with zero integration error the two trailing edge points will never match up. You can maybe apply a transformation to get them lined up, but that may skew the geometry. I was never curious enough to find a proper solution to this coordinate coincidence problem as other more obvious smoothing methods generally gave acceptable results.

I would say vector calculus, functions with more than one variables their limits. derivatives, integrals and applications

May I ask what topics calculus 3 normally covers? In my university, there is no Calculus 3 course. We learn multivariable calculus in one course, and then differential geometry (curvature and other stuff) in another course etc.

Curvature is the second derivative of position with respect to arclength. In other words, it's the acceleration if you parametrize the curve by arclength. (But what I'm describing is a vector; the curvature in this video is its magnitude.)

probably you can find the biggest curvature value of e^x by finding its curvature function, taking the derivative, and making equal to 0. (finding the maximum of the curvature function)

Hey bprp! I saw your previous video on curvature where there is an easy formula for the curvature with the given parametric equations. I noticed that formula only works when there are 2 components. Is there a similar formula for the curvature when there are 3 components?

Does it matter which is the parametrization or is it like the line integral?

Maybe you could show an example using parabolic coordinate or the Frenet base?

yay eighth one!!

i am late to the party . "insert sad emoji "