Find the curvature K of the curve at the point P. K = r(t) = et cos(t)i + et sin(t)j + e¹k, P(1, 0, 1)
Find the curvature K of the curve at the point P. K = r(t) = et cos(t)i + et sin(t)j + e¹k, P(1, 0, 1)
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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The answer was not what I found from another answer on here which said it was 0.17 or in fraction form sqr root of 2/9 * e^-1.
![**Finding the Curvature of a Curve at a Given Point**
**Problem Statement:**
Find the curvature \( K \) of the curve at the point \( P \).
Given the parametric equations of the curve:
\[ \mathbf{r}(t) = e^t \cos(t)\mathbf{i} + e^t \sin(t)\mathbf{j} + e^t\mathbf{k}, \quad P(1, 0, 1) \]
**Curvature Calculation:**
\[ K = \boxed{\quad} \]
The provided equations represent a parametric form of a space curve where:
- \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the standard unit vectors in the x, y, and z directions respectively.
- \( t \) is a parameter that influences the position on the curve.
To find the curvature \( K \) at the given point, you would need to evaluate the magnitude of the curvature vector typically defined using the derivatives of the position vector with respect to \( t \). Note that the specific numeric value is to be determined as part of solving the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c3592c0-3a06-4a30-a95f-c9eb1d9570de%2F416905a1-7a05-4501-bd27-c9010ca59c89%2Fzrvx1oe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Curvature of a Curve at a Given Point**
**Problem Statement:**
Find the curvature \( K \) of the curve at the point \( P \).
Given the parametric equations of the curve:
\[ \mathbf{r}(t) = e^t \cos(t)\mathbf{i} + e^t \sin(t)\mathbf{j} + e^t\mathbf{k}, \quad P(1, 0, 1) \]
**Curvature Calculation:**
\[ K = \boxed{\quad} \]
The provided equations represent a parametric form of a space curve where:
- \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the standard unit vectors in the x, y, and z directions respectively.
- \( t \) is a parameter that influences the position on the curve.
To find the curvature \( K \) at the given point, you would need to evaluate the magnitude of the curvature vector typically defined using the derivatives of the position vector with respect to \( t \). Note that the specific numeric value is to be determined as part of solving the problem.
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