I think you should restate the last part. The one where u represents position. The derivative of u represents velocity not acceleration. Position is a vector with its origin on the center of the circle and velocity (u') is a vector tangential to the circle thus orthogonal to u. It can vary in length but is always orthogonal to the position vector.
Whats interesting to me is the subtle transitions between ordinary calculus, and vector differentiation. You start with an equality uu=c which contains vectors, but which is just an equation between scalar functions of t, you apply ordinary differentiation, and you somehow end up with differentiated vector functions. Its like the product rule almost "translates" between the two! I'd love a more in-depth exploration of this, even just to make it more intuitive
And it is necessarily true in a linear space but not necessarily true in Euclidean space. Something you see in physics papers are references to 3+1 dimensions since the parameter t is purely abstract. I.e., "perpendicular" is observable but "linear independence" *can be* but might not be. Or put another way: how do you distinguish "linear independent" from "perpendicular". The distinction isn't always important, but it is one that Poincare had a problem with in some discussion he had related to Einstein's model of gravity. If you want a citation, I'll look it up. BTW, I'm not criticizing, just adding. I realize you're showing the elegance of vector based reasoning and I agree. It is! Your teaching style is too. I've read ta book titled "Elementary General Relativity" and compared to it, your stuff is awesome! Maybe this time I'll be able to get past "dual metrics".
🤯You know that time when you said to yourself, "OK, I understand this now" and then later 🤯oh NOW I understand it ... um ...🤯 I already knew this. Probably. I mean clearly. Obviously. I just never thought of it before.
I think you should restate the last part. The one where u represents position. The derivative of u represents velocity not acceleration. Position is a vector with its origin on the center of the circle and velocity (u') is a vector tangential to the circle thus orthogonal to u. It can vary in length but is always orthogonal to the position vector.
Whats interesting to me is the subtle transitions between ordinary calculus, and vector differentiation. You start with an equality uu=c which contains vectors, but which is just an equation between scalar functions of t, you apply ordinary differentiation, and you somehow end up with differentiated vector functions. Its like the product rule almost "translates" between the two! I'd love a more in-depth exploration of this, even just to make it more intuitive
Yes, it's kinda magical! I don't know if I have a more in-depth explanation, but more examples are coming.
Thank you for your kind comment!
And it is necessarily true in a linear space but not necessarily true in Euclidean space. Something you see in physics papers are references to 3+1 dimensions since the parameter t is purely abstract. I.e., "perpendicular" is observable but "linear independence" *can be* but might not be. Or put another way: how do you distinguish "linear independent" from "perpendicular". The distinction isn't always important, but it is one that Poincare had a problem with in some discussion he had related to Einstein's model of gravity. If you want a citation, I'll look it up.
BTW, I'm not criticizing, just adding. I realize you're showing the elegance of vector based reasoning and I agree. It is! Your teaching style is too. I've read ta book titled "Elementary General Relativity" and compared to it, your stuff is awesome! Maybe this time I'll be able to get past "dual metrics".
🤯You know that time when you said to yourself, "OK, I understand this now" and then later 🤯oh NOW I understand it ... um ...🤯
I already knew this. Probably. I mean clearly. Obviously. I just never thought of it before.
👍
Stokes' theorem when?
Not sure, but definitely coming!