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**Problem 2:** Let \(\vec{A} = x^3y \, \hat{a}_x - y^2 \, \hat{a}_y\), and let \(L\) be a contour as shown. a. Find \(\nabla \times \vec{A}\). b. Find \(\oint_L \vec{A} \cdot d\vec{l}\). c. Find \(\iint_S \nabla \times \vec{A} \cdot d\vec{S}\) where \(S\) is the area bounded by \(L\). d. Show that Stokes's Theorem is satisfied. e. Find \(\nabla \cdot (\nabla \times \vec{A})\). **Diagram Explanation:** The diagram is a triangular representation in the xy-plane. It shows a triangular area with vertices at \((0,0)\), \((1,0)\), and \((0,1)\). The arrows indicate that the contour \(L\) follows the edges of the triangle. The arrow directions suggest an anti-clockwise traversal around the triangle's vertices. The area \(S\) mentioned in the problem is the area enclosed within this triangular contour.

**Problem 2:** Let \(\vec{A} = x^3y \, \hat{a}_x - y^2 \, \hat{a}_y\), and let \(L\) be a contour as shown. a. Find \(\nabla \times \vec{A}\). b. Find \(\oint_L \vec{A} \cdot d\vec{l}\). c. Find \(\iint_S \nabla \times \vec{A} \cdot d\vec{S}\) where \(S\) is the area bounded by \(L\). d. Show that Stokes's Theorem is satisfied. e. Find \(\nabla \cdot (\nabla \times \vec{A})\). **Diagram Explanation:** The diagram is a triangular representation in the xy-plane. It shows a triangular area with vertices at \((0,0)\), \((1,0)\), and \((0,1)\). The arrows indicate that the contour \(L\) follows the edges of the triangle. The arrow directions suggest an anti-clockwise traversal around the triangle's vertices. The area \(S\) mentioned in the problem is the area enclosed within this triangular contour.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Problem 2:** Let \(\vec{A} = x^3y \, \hat{a}_x - y^2 \, \hat{a}_y\), and let \(L\) be a contour as shown. a. Find \(\nabla \times \vec{A}\). b. Find \(\oint_L \vec{A} \cdot d\vec{l}\). c. Find \(\iint_S \nabla \times \vec{A} \cdot d\vec{S}\) where \(S\) is the area bounded by \(L\). d. Show that Stokes's Theorem is satisfied. e. Find \(\nabla \cdot (\nabla \times \vec{A})\). **Diagram Explanation:** The diagram is a triangular representation in the xy-plane. It shows a triangular area with vertices at \((0,0)\), \((1,0)\), and \((0,1)\). The arrows indicate that the contour \(L\) follows the edges of the triangle. The arrow directions suggest an anti-clockwise traversal around the triangle's vertices. The area \(S\) mentioned in the problem is the area enclosed within this triangular contour.
Transcribed Image Text:**Problem 2:** Let \(\vec{A} = x^3y \, \hat{a}_x - y^2 \, \hat{a}_y\), and let \(L\) be a contour as shown. a. Find \(\nabla \times \vec{A}\). b. Find \(\oint_L \vec{A} \cdot d\vec{l}\). c. Find \(\iint_S \nabla \times \vec{A} \cdot d\vec{S}\) where \(S\) is the area bounded by \(L\). d. Show that Stokes's Theorem is satisfied. e. Find \(\nabla \cdot (\nabla \times \vec{A})\). **Diagram Explanation:** The diagram is a triangular representation in the xy-plane. It shows a triangular area with vertices at \((0,0)\), \((1,0)\), and \((0,1)\). The arrows indicate that the contour \(L\) follows the edges of the triangle. The arrow directions suggest an anti-clockwise traversal around the triangle's vertices. The area \(S\) mentioned in the problem is the area enclosed within this triangular contour.
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