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**Exercise:** Let \(\vec{A} = x^3 \vec{a}_x - y^2 \vec{a}_y\), and let \(L\) be a contour as shown. **Diagram:** The diagram consists of a right triangle in the \(xy\)-plane with vertices at the origin \((0, 0)\), \((1, 0)\) on the \(x\)-axis, and \((0, 1)\) on the \(y\)-axis. The line segments form a closed loop which is traversed counterclockwise. The area enclosed by the triangle is denoted as \(S\). **Tasks:** 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 \(\nabla \cdot (\nabla \times \vec{A})\). This set of exercises helps in verifying Stokes's theorem and understanding the concepts of curl and divergence in vector calculus.

**Exercise:** Let \(\vec{A} = x^3 \vec{a}_x - y^2 \vec{a}_y\), and let \(L\) be a contour as shown. **Diagram:** The diagram consists of a right triangle in the \(xy\)-plane with vertices at the origin \((0, 0)\), \((1, 0)\) on the \(x\)-axis, and \((0, 1)\) on the \(y\)-axis. The line segments form a closed loop which is traversed counterclockwise. The area enclosed by the triangle is denoted as \(S\). **Tasks:** 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 \(\nabla \cdot (\nabla \times \vec{A})\). This set of exercises helps in verifying Stokes's theorem and understanding the concepts of curl and divergence in vector calculus.

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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**Exercise:** Let \(\vec{A} = x^3 \vec{a}_x - y^2 \vec{a}_y\), and let \(L\) be a contour as shown. **Diagram:** The diagram consists of a right triangle in the \(xy\)-plane with vertices at the origin \((0, 0)\), \((1, 0)\) on the \(x\)-axis, and \((0, 1)\) on the \(y\)-axis. The line segments form a closed loop which is traversed counterclockwise. The area enclosed by the triangle is denoted as \(S\). **Tasks:** 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 \(\nabla \cdot (\nabla \times \vec{A})\). This set of exercises helps in verifying Stokes's theorem and understanding the concepts of curl and divergence in vector calculus.
Transcribed Image Text:**Exercise:** Let \(\vec{A} = x^3 \vec{a}_x - y^2 \vec{a}_y\), and let \(L\) be a contour as shown. **Diagram:** The diagram consists of a right triangle in the \(xy\)-plane with vertices at the origin \((0, 0)\), \((1, 0)\) on the \(x\)-axis, and \((0, 1)\) on the \(y\)-axis. The line segments form a closed loop which is traversed counterclockwise. The area enclosed by the triangle is denoted as \(S\). **Tasks:** 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 \(\nabla \cdot (\nabla \times \vec{A})\). This set of exercises helps in verifying Stokes's theorem and understanding the concepts of curl and divergence in vector calculus.
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