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Let C2 be the boundary of the triangle whose vertices are (1, 2), (-3, 1), and (-2, 2). Suppose C2 is oriented clockwise. Let G(x, y) = (4x cos(x - y) — 2x² sin(x - y), 2x² sin(x − y) + 4x). √ G. dR. Use Green's Theorem to evaluate

Let C2 be the boundary of the triangle whose vertices are (1, 2), (-3, 1), and (-2, 2). Suppose C2 is oriented clockwise. Let G(x, y) = (4x cos(x - y) — 2x² sin(x - y), 2x² sin(x − y) + 4x). √ G. dR. Use Green's Theorem to evaluate

Advanced Engineering Mathematics
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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3.7 Green's Theorem In this section, we will see that a line integral along a simple closed plane curve C is related to an ordinary double integral over the plane region R with boundary C. The boundary curve C is said to be positively oriented if it is equipped with a parametrization R(t) such that the region R remains on the left as R(t) traces the curve C. The symbol F-dR=Pd. $ then denotes a line integral around C with positive orientation. The following result was published by George Green in 1828. Theorem 3.7.1 (Green's Theorem). Let C be a positively oriented piecewise-smooth simple closed curve that bounds the region R in the xy-plane. Suppose that P and Q are scalar fields of x and Y with continuous first-order partial derivatives on R. Then 'მი & F Pdx + Qdy = f( R ?х - ӘР ду dA.
Transcribed Image Text:3.7 Green's Theorem In this section, we will see that a line integral along a simple closed plane curve C is related to an ordinary double integral over the plane region R with boundary C. The boundary curve C is said to be positively oriented if it is equipped with a parametrization R(t) such that the region R remains on the left as R(t) traces the curve C. The symbol F-dR=Pd. $ then denotes a line integral around C with positive orientation. The following result was published by George Green in 1828. Theorem 3.7.1 (Green's Theorem). Let C be a positively oriented piecewise-smooth simple closed curve that bounds the region R in the xy-plane. Suppose that P and Q are scalar fields of x and Y with continuous first-order partial derivatives on R. Then 'მი & F Pdx + Qdy = f( R ?х - ӘР ду dA.
Let C₂ be the boundary of the triangle whose vertices are (1, 2), (-3, 1), and (−2, 2). Suppose C2 is oriented clockwise. Let G(x, y) = (4x cos(x - y) — 2x² sin(x - y), 2x² sin(x − y) + 4x). - Use Green's Theorem to evaluate Jc₂ G.dR.
Transcribed Image Text:Let C₂ be the boundary of the triangle whose vertices are (1, 2), (-3, 1), and (−2, 2). Suppose C2 is oriented clockwise. Let G(x, y) = (4x cos(x - y) — 2x² sin(x - y), 2x² sin(x − y) + 4x). - Use Green's Theorem to evaluate Jc₂ G.dR.
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