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EXAMPLE 1 Evaluate ſ F. dr, where F(x, y, z) = -y2i+ 3xj + 5z2k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the fiqure. Although [F. dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute D i k a ax ay az Video Example4) curl F = = ( )k 3x 5z2 Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y +z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x2 + y2 < 1 and so with z = g(x, y) = 2 - y, we have SF· dr = curl F· dS = dA (3 + )r dr de 3r2 Jó de %3D 2 2 3 (-+ )de -(2x) +0 = 37 Read It

EXAMPLE 1 Evaluate ſ F. dr, where F(x, y, z) = -y2i+ 3xj + 5z2k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the fiqure. Although [F. dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute D i k a ax ay az Video Example4) curl F = = ( )k 3x 5z2 Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y +z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x2 + y2 < 1 and so with z = g(x, y) = 2 - y, we have SF· dr = curl F· dS = dA (3 + )r dr de 3r2 Jó de %3D 2 2 3 (-+ )de -(2x) +0 = 37 Read It

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.2: Ellipses
Problem 34E
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Do please Frgh 16.8
EXAMPLE 1 Evaluate ſ F. dr, where F(x, y, z) = -y?i + 3xj + 5z?k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the figure. Although SF · dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute D i a ax ay az Video Example4) curl F = = ( )k 3x 5z2 Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x2 + y2 < 1 and so with z = g(x, y) = 2 - y, we have SF· dr = curl F. dS = dA (3 + )r dr de - L 3r2 Jó de %3D 2 2 3 (-+ )de -(2x) + 0 = 37 Read It
Transcribed Image Text:EXAMPLE 1 Evaluate ſ F. dr, where F(x, y, z) = -y?i + 3xj + 5z?k and C is the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1. (Orient C to be counterclockwise when viewed from above.) SOLUTION The curve C (an ellipse) is shown in the figure. Although SF · dr could be evaluated directly, it's easier to use Stokes' Theorem. We first compute D i a ax ay az Video Example4) curl F = = ( )k 3x 5z2 Although there are many surfaces with boundary C, the most convenient choice is the elliptical region S in the plane y + z = 2 that is bounded by C. If we orient S upwards, then C has the induced positive orientation. The projection D of S on the xy-plane is the disk x2 + y2 < 1 and so with z = g(x, y) = 2 - y, we have SF· dr = curl F. dS = dA (3 + )r dr de - L 3r2 Jó de %3D 2 2 3 (-+ )de -(2x) + 0 = 37 Read It
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