A circle C in the plane x+y+z=4 has a radius of 3 and center (3, - 1,2). Evaluate O F.dr for F= (0,2z, - y), where C has counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the circle? Does it depend on the location of the center of the circle?
A circle C in the plane x+y+z=4 has a radius of 3 and center (3, - 1,2). Evaluate O F.dr for F= (0,2z, - y), where C has counterclockwise orientation when viewed from above. Does the circulation depend on the radius of the circle? Does it depend on the location of the center of the circle?
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![### Circulation of a Circle in the Plane x + y + z = 4
A circle \(C\) in the plane \(x + y + z = 4\) has a radius of 3 and center \((3, -1, 2)\). Evaluate \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\) for \(\mathbf{F} = \langle 0, 2z, -y \rangle\), where \(C\) has counterclockwise orientation when viewed from above.
**Questions:**
1. Does the circulation depend on the radius of the circle?
2. Does it depend on the location of the center of the circle?
**Integral to evaluate:**
\[
\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \boxed{ }
\]
(*Type an exact answer, using \(\pi\) as needed*)
**Does the circulation depend on the radius of the circle?**
- \( \circ \) A. Yes. The circulation is inversely proportional to the radius.
- \( \circ \) B. Yes. The circulation is proportional to the radius.
- \( \circ \) C. Yes. The circulation is proportional to the square of the radius.
- \( \circ \) D. No. The circulation does not change with the radius.
- \( \circ \) E. Yes. The circulation is inversely proportional to the square of the radius.
**Does the circulation depend on the location of the center of the circle?**
- \( \circ \) A. No. The circulation does not change when the circle moves in the plane.
- \( \circ \) B. Yes. The circulation has a complex relationship with the location of the circle.
- \( \circ \) C. Yes. The circulation changes linearly with \(y\).
- \( \circ \) D. Yes. The circulation changes linearly with \(z\).
- \( \circ \) E. Yes. The circulation changes linearly with \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fdb95a5-48dd-4f99-85af-89f8f2873eee%2Fff5cf896-db5f-486e-b118-60203c16f2b9%2F5u5z4l.jpeg&w=3840&q=75)
Transcribed Image Text:### Circulation of a Circle in the Plane x + y + z = 4
A circle \(C\) in the plane \(x + y + z = 4\) has a radius of 3 and center \((3, -1, 2)\). Evaluate \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\) for \(\mathbf{F} = \langle 0, 2z, -y \rangle\), where \(C\) has counterclockwise orientation when viewed from above.
**Questions:**
1. Does the circulation depend on the radius of the circle?
2. Does it depend on the location of the center of the circle?
**Integral to evaluate:**
\[
\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \boxed{ }
\]
(*Type an exact answer, using \(\pi\) as needed*)
**Does the circulation depend on the radius of the circle?**
- \( \circ \) A. Yes. The circulation is inversely proportional to the radius.
- \( \circ \) B. Yes. The circulation is proportional to the radius.
- \( \circ \) C. Yes. The circulation is proportional to the square of the radius.
- \( \circ \) D. No. The circulation does not change with the radius.
- \( \circ \) E. Yes. The circulation is inversely proportional to the square of the radius.
**Does the circulation depend on the location of the center of the circle?**
- \( \circ \) A. No. The circulation does not change when the circle moves in the plane.
- \( \circ \) B. Yes. The circulation has a complex relationship with the location of the circle.
- \( \circ \) C. Yes. The circulation changes linearly with \(y\).
- \( \circ \) D. Yes. The circulation changes linearly with \(z\).
- \( \circ \) E. Yes. The circulation changes linearly with \(x\).
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