The curve C is the circle of radius 7 contained in the plane z = 7 and centered around the z-axis, oriented clockwise when viewed from above. (In(x¹ + 1) − 2y, e³² + 5z, 17x If F(x, y, z) lo = F. dr = cos (2³)) then:
The curve C is the circle of radius 7 contained in the plane z = 7 and centered around the z-axis, oriented clockwise when viewed from above. (In(x¹ + 1) − 2y, e³² + 5z, 17x If F(x, y, z) lo = F. dr = cos (2³)) then:
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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![## Problem Statement
The curve \( C \) is the circle of radius 7 contained in the plane \( z = 7 \) and centered around the \( z \)-axis, oriented clockwise when viewed from above.
If \(\mathbf{F}(x, y, z) = \left\langle \ln(x^4 + 1) - 2y, e^{y^2} + 5z, 17x - \cos(z^3) \right\rangle\), then:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \boxed{\phantom{ }}
\]
### Explanation
This expression denotes a line integral over the curve \( C \) for the vector field \( \mathbf{F}(x, y, z) \). The line integral represented is \(\oint_C \mathbf{F} \cdot d\mathbf{r}\), where the integration path is the circle \( C \) in the described plane.
The given vector field \(\mathbf{F}(x, y, z)\) consists of three components:
- \( \ln(x^4 + 1) - 2y \) in the x-direction,
- \( e^{y^2} + 5z \) in the y-direction,
- \( 17x - \cos(z^3) \) in the z-direction.
The problem requires evaluating this integral, taking into account the orientation and position of the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2Fe28e2dc7-1f06-4378-94db-943024caa35e%2Fpdab5ao_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
The curve \( C \) is the circle of radius 7 contained in the plane \( z = 7 \) and centered around the \( z \)-axis, oriented clockwise when viewed from above.
If \(\mathbf{F}(x, y, z) = \left\langle \ln(x^4 + 1) - 2y, e^{y^2} + 5z, 17x - \cos(z^3) \right\rangle\), then:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \boxed{\phantom{ }}
\]
### Explanation
This expression denotes a line integral over the curve \( C \) for the vector field \( \mathbf{F}(x, y, z) \). The line integral represented is \(\oint_C \mathbf{F} \cdot d\mathbf{r}\), where the integration path is the circle \( C \) in the described plane.
The given vector field \(\mathbf{F}(x, y, z)\) consists of three components:
- \( \ln(x^4 + 1) - 2y \) in the x-direction,
- \( e^{y^2} + 5z \) in the y-direction,
- \( 17x - \cos(z^3) \) in the z-direction.
The problem requires evaluating this integral, taking into account the orientation and position of the curve.
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