- #1
Cyrus
- 3,238
- 16
A quick question. In stewart, the curvature is defined as:
[tex] \kappa = \abs(\frac{dT}{ds}) [/tex]
and it says that:
"The curvature is easier to compute if it is expressed in terms of the parameter t instead of s, so we can use the Chain Rule to write:
[tex] \frac{dT}{dt}=\frac{dT}{ds} \frac{ds}{dt} [/tex]
and
[tex] \kappa= \abs(\frac{dT}{ds}) = \abs(\frac{dT/dt}{ds/dt}) [/tex].
I don't see how they used the chain rule on that problem, normally, it is T(t).
So T is a function of time, (t) directly. Are they saying that they replaced the variable t, in terms of the varaible s. S, which is arclength, is a function of t. So I suppose the rewrote t, as a function of s instead. The same as saying, [tex] y=x^2 [/tex] or you could also say, [tex] x=+/-\sqrt(y) [/tex].
And they pluged this in where (t) used to be in T(t). So now it is written as:
T(s), but s(t), so it is like saying T(s(t)) ?
Also, did they just solve the first fraction dT/dt =dT/ds ds/dt by dividing both sides by ds/dt in order to get, dT/ds= (dT/dt)/(ds/dt). Is it correct to multiple and divide by ds/dt like that?
so when you do the chain rule you get, dT/dt = dT/ds * ds/dt ?
On another note, I am taking dynamics now, and it is quite amazing how I looked at three books, physics 1, dynamics, and calc 3, and found three different ways of finding the acceleration vector, the dynamics and physics book do almost the same approach with diagrams of vectors, and the math book quite nicely does the same thing without every using a picture. Neat, I think.
[tex] \kappa = \abs(\frac{dT}{ds}) [/tex]
and it says that:
"The curvature is easier to compute if it is expressed in terms of the parameter t instead of s, so we can use the Chain Rule to write:
[tex] \frac{dT}{dt}=\frac{dT}{ds} \frac{ds}{dt} [/tex]
and
[tex] \kappa= \abs(\frac{dT}{ds}) = \abs(\frac{dT/dt}{ds/dt}) [/tex].
I don't see how they used the chain rule on that problem, normally, it is T(t).
So T is a function of time, (t) directly. Are they saying that they replaced the variable t, in terms of the varaible s. S, which is arclength, is a function of t. So I suppose the rewrote t, as a function of s instead. The same as saying, [tex] y=x^2 [/tex] or you could also say, [tex] x=+/-\sqrt(y) [/tex].
And they pluged this in where (t) used to be in T(t). So now it is written as:
T(s), but s(t), so it is like saying T(s(t)) ?
Also, did they just solve the first fraction dT/dt =dT/ds ds/dt by dividing both sides by ds/dt in order to get, dT/ds= (dT/dt)/(ds/dt). Is it correct to multiple and divide by ds/dt like that?
so when you do the chain rule you get, dT/dt = dT/ds * ds/dt ?
On another note, I am taking dynamics now, and it is quite amazing how I looked at three books, physics 1, dynamics, and calc 3, and found three different ways of finding the acceleration vector, the dynamics and physics book do almost the same approach with diagrams of vectors, and the math book quite nicely does the same thing without every using a picture. Neat, I think.
Last edited: