Let C be the curve r(t) = (2t, t², ť /3). (a) Compute the Frenet frame for C. (b) Calculate the values of T(t), N(t), and B(t) as t → ±00. (c) Compute the curvature of C.
Let C be the curve r(t) = (2t, t², ť /3). (a) Compute the Frenet frame for C. (b) Calculate the values of T(t), N(t), and B(t) as t → ±00. (c) Compute the curvature of C.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![3. Let \( C \) be the curve \(\mathbf{r}(t) = \langle 2t, t^2, t^3/3 \rangle\).
(a) Compute the Frenet frame for \( C \).
(b) Calculate the values of \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) as \( t \to \pm \infty \).
(c) Compute the curvature of \( C \).
(d) The torsion \(\tau(t)\) of a curve with position vector \(\mathbf{r}(t)\) is given by
\[
\tau(t) = \frac{(\mathbf{r}'(t) \times \mathbf{r}''(t)) \cdot \mathbf{r}'''(t)}{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|^2}
\]
Compute the torsion of the curve \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd93ae1f0-5948-4075-83b0-747f4fbdf146%2Fcf9012b7-5cf9-4ce8-b65a-69fae441bc02%2Fnq1xcm_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let \( C \) be the curve \(\mathbf{r}(t) = \langle 2t, t^2, t^3/3 \rangle\).
(a) Compute the Frenet frame for \( C \).
(b) Calculate the values of \(\mathbf{T}(t)\), \(\mathbf{N}(t)\), and \(\mathbf{B}(t)\) as \( t \to \pm \infty \).
(c) Compute the curvature of \( C \).
(d) The torsion \(\tau(t)\) of a curve with position vector \(\mathbf{r}(t)\) is given by
\[
\tau(t) = \frac{(\mathbf{r}'(t) \times \mathbf{r}''(t)) \cdot \mathbf{r}'''(t)}{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|^2}
\]
Compute the torsion of the curve \( C \).
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