5 Let fi R²→R be the vector field given by F (x,y)- (4,-x). A- Calculate the circulation of f when it travels along the at the ongin, and and C=(2,0o) with the oniented from A to C. semicircumference C of radus 2 centered Joins pointa A= (-2,0), B = (0,2) %3D B- Calculate the integral over C, the Curve from A of scalar field 3: R→R 9iven by
5 Let fi R²→R be the vector field given by F (x,y)- (4,-x). A- Calculate the circulation of f when it travels along the at the ongin, and and C=(2,0o) with the oniented from A to C. semicircumference C of radus 2 centered Joins pointa A= (-2,0), B = (0,2) %3D B- Calculate the integral over C, the Curve from A of scalar field 3: R→R 9iven by
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem 5:**
Let \(\vec{f} : \mathbb{R}^2 \to \mathbb{R}\) be the vector field given by \(\vec{f}(x, y) = (y, -x)\).
**A)** Calculate the circulation of \(\vec{f}\) when it travels along the semicircumference \(C\) of radius 2 centered at the origin, and joins points \(A = (-2, 0)\), \(B = (0, 2)\), and \(C = (2, 0)\) with the orientation from \(A\) to \(C\).
**B)** Calculate the integral over \(C\), the curve from \(A\) of the scalar field \(\vec{g} : \mathbb{R}^2 \to \mathbb{R}\) given by
\[ g(x, y) = y \left\| \vec{f}(x, y) \right\| \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d456929-ce4c-4399-bf9c-61ec96afc9f0%2F65a6c887-7924-4ce5-a84a-1e8757a3ed89%2Flizde6r_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
Let \(\vec{f} : \mathbb{R}^2 \to \mathbb{R}\) be the vector field given by \(\vec{f}(x, y) = (y, -x)\).
**A)** Calculate the circulation of \(\vec{f}\) when it travels along the semicircumference \(C\) of radius 2 centered at the origin, and joins points \(A = (-2, 0)\), \(B = (0, 2)\), and \(C = (2, 0)\) with the orientation from \(A\) to \(C\).
**B)** Calculate the integral over \(C\), the curve from \(A\) of the scalar field \(\vec{g} : \mathbb{R}^2 \to \mathbb{R}\) given by
\[ g(x, y) = y \left\| \vec{f}(x, y) \right\| \]
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