2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = r² and 2 = r* (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to ř '(t) and F"(t). In fact, an alternate formula for this vector is F'(t) × F"(t) B(t): F'(t) × "(t)|
2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = r² and 2 = r* (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to ř '(t) and F"(t). In fact, an alternate formula for this vector is F'(t) × F"(t) B(t): F'(t) × "(t)|
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![2. Complete all parts:
(a) Find the equation of the curve of intersection of the surfaces \( y = x^2 \) and \( z = x^3 \).
(b) What is the name of the resulting curve of intersection?
(c) Find the equation for \(\mathbf{B}\), the unit binormal vector to the curve when \( t = 1 \).
Hint: Instead of using the usual formula for \(\mathbf{B}\), note that the unit binormal vector is orthogonal to \(\mathbf{r}'(t)\) and \(\mathbf{r}''(t)\). In fact, an alternate formula for this vector is
\[
\mathbf{B}(t) = \frac{\mathbf{r}'(t) \times \mathbf{r}''(t)}{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9efe910-fb5f-41b2-a82d-eda9e45aba44%2F60a1932f-0ba0-456e-944c-10d049cd3167%2Fj20ym3l_processed.png&w=3840&q=75)
Transcribed Image Text:2. Complete all parts:
(a) Find the equation of the curve of intersection of the surfaces \( y = x^2 \) and \( z = x^3 \).
(b) What is the name of the resulting curve of intersection?
(c) Find the equation for \(\mathbf{B}\), the unit binormal vector to the curve when \( t = 1 \).
Hint: Instead of using the usual formula for \(\mathbf{B}\), note that the unit binormal vector is orthogonal to \(\mathbf{r}'(t)\) and \(\mathbf{r}''(t)\). In fact, an alternate formula for this vector is
\[
\mathbf{B}(t) = \frac{\mathbf{r}'(t) \times \mathbf{r}''(t)}{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}
\]
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