I think its important to clarify that dT/ds is not a 'rate change' as it doesnt change wrt to time. It changes in proportion to s, essentially having the effect of 'normalising' the varying speeds at which a parameterization can travel along a path. e.g. (cos(e^t), sin(e^t)) is not distorted so to speak.
I like to think of it as the ratio of time dependent rates. It makes it easier when you start to talk about normal/tangential coordinate systems and acceleration. The normal component of acceleration winds up affecting the rate of change of direction wrt time (the centripetal acceleration), the tangential component of acceleration affects the rate of change of the distance traveled along the path wrt time (speed)
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.
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.)
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.
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?
dT/ds - rate of change of direction. ds/dt - rate of change of distance traveled along the path
I think its important to clarify that dT/ds is not a 'rate change' as it doesnt change wrt to time. It changes in proportion to s, essentially having the effect of 'normalising' the varying speeds at which a parameterization can travel along a path. e.g. (cos(e^t), sin(e^t)) is not distorted so to speak.
I like to think of it as the ratio of time dependent rates. It makes it easier when you start to talk about normal/tangential coordinate systems and acceleration. The normal component of acceleration winds up affecting the rate of change of direction wrt time (the centripetal acceleration), the tangential component of acceleration affects the rate of change of the distance traveled along the path wrt time (speed)
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.
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.)
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.
A lot of it involves calculus of multi variable functions and vectors.
I would say vector calculus, functions with more than one variables their limits. derivatives, integrals and applications
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)
Maybe you could show an example using parabolic coordinate or the Frenet base?
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?
Yes. Here are all the helpful formulas for curvature ruclips.net/video/41EtLvQrXu4/видео.html
Does it matter which is the parametrization or is it like the line integral?
yay eighth one!!
i am late to the party . "insert sad emoji "