Use Green's Theorem to calculate the circulation of F = 3xyi around the rectangle 0 ≤ x ≤ 2,0 ≤ y ≤ 3 oriented counterclockwise. , circulation =
Use Green's Theorem to calculate the circulation of F = 3xyi around the rectangle 0 ≤ x ≤ 2,0 ≤ y ≤ 3 oriented counterclockwise. , circulation =
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
![**Title: Calculating Circulation Using Green's Theorem**
**Objective:**
To calculate the circulation of the vector field \(\vec{F} = 3xy \vec{i}\) around the rectangle defined by \(0 \leq x \leq 2, 0 \leq y \leq 3\), oriented counterclockwise.
**Instructions:**
Use Green's Theorem, which relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(D\) bounded by \(C\).
**Formula:**
Green's Theorem can be stated as:
\[
\oint_C \vec{F} \cdot d\vec{r} = \iint_D \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA
\]
where \(\vec{F} = M\vec{i} + N\vec{j}\).
For \(\vec{F} = 3xy\vec{i}\), we have:
- \(M = 3xy\)
- \(N = 0\)
To find the circulation, we calculate:
\[
\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 - 3x
\]
**Calculation Domain:**
Integrate over the rectangle defined by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\).
**Solution:**
Evaluate the double integral:
\[
\iint_D -3x \, dA = \int_0^3 \int_0^2 (-3x) \, dx \, dy
\]
Perform the integration, and calculate the circulation.
**Conclusion:**
Fill in the calculated circulation:
\[
\text{circulation} = \boxed{}
\]
**Notes:**
- Ensure calculations are accurate.
- Remember that the vector field has only one nonzero component.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff35f8f89-65fc-4b90-98e9-68f8eaa1b3da%2F55a4da15-3699-4d93-94b7-5e265037f1e7%2Fzfpr7lq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Circulation Using Green's Theorem**
**Objective:**
To calculate the circulation of the vector field \(\vec{F} = 3xy \vec{i}\) around the rectangle defined by \(0 \leq x \leq 2, 0 \leq y \leq 3\), oriented counterclockwise.
**Instructions:**
Use Green's Theorem, which relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(D\) bounded by \(C\).
**Formula:**
Green's Theorem can be stated as:
\[
\oint_C \vec{F} \cdot d\vec{r} = \iint_D \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA
\]
where \(\vec{F} = M\vec{i} + N\vec{j}\).
For \(\vec{F} = 3xy\vec{i}\), we have:
- \(M = 3xy\)
- \(N = 0\)
To find the circulation, we calculate:
\[
\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 - 3x
\]
**Calculation Domain:**
Integrate over the rectangle defined by \(0 \leq x \leq 2\) and \(0 \leq y \leq 3\).
**Solution:**
Evaluate the double integral:
\[
\iint_D -3x \, dA = \int_0^3 \int_0^2 (-3x) \, dx \, dy
\]
Perform the integration, and calculate the circulation.
**Conclusion:**
Fill in the calculated circulation:
\[
\text{circulation} = \boxed{}
\]
**Notes:**
- Ensure calculations are accurate.
- Remember that the vector field has only one nonzero component.
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