The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: 1. (dT)/(ds) = κN 2. (dN)/(ds) = -κT τB 3. (dB)/(ds) = -τN (Formula 1 comes from Exercise 59 and Formula 3 comes from Exercise 61.) Use the fact that N = B ×T to deduce Formula 2 from Formulas 1 and 3. | Numerade
The following formulas, called the Frenet-Serret formulas, are of funda…
26 feb. 2021
numerade.com
The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: 1. d 𝐓 / d s=κ𝐍 2. d 𝐍 / d s=-κ𝐓 τ𝐁 3. d 𝐁 / d s=-τ𝐍 (Formula 1 comes from Exercise 63 and Formula 3 is Equation 12.) Use the Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with respect to t . Start as in the proof of Theorem 10 . ) (a) 𝐫^''=s^'' 𝐓 κ(s^')^2 𝐍 (b) 𝐫^' ×𝐫^''=κ(s^')^3 𝐁 (c) 𝐫^'''=[s^'''-κ^2(s^')^3] 𝐓 [3 κs^' s^'' κ^'(s
The following formulas, called the Frenet-Serret formulas, are of funda…
23 mei 2021
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