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EXAMPLE 5 Show that the vector field F(x, y, z) = xz²i + x?yz®j - y*k can't be written as the curl of another vector field, that is, F + curl G. SOLUTION As we showed in this example, it can be shown that div F = and therefore div F = v 0. If it were true that F = curl G, then this theorem would give div F = div curl G = 0 which contradicts div F = v0. Therefore F is not the curl of another vector field.

EXAMPLE 5 Show that the vector field F(x, y, z) = xz²i + x?yz®j - y*k can't be written as the curl of another vector field, that is, F + curl G. SOLUTION As we showed in this example, it can be shown that div F = and therefore div F = v 0. If it were true that F = curl G, then this theorem would give div F = div curl G = 0 which contradicts div F = v0. Therefore F is not the curl of another vector field.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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EXAMPLE 5 Show that the vector field F(x, y, z) = xz'i + x'yz°j - yk can't be written as the curl of another vector field, that is, F * curl G. SOLUTION As we showed in this example, it can be shown that div F = and therefore div F = v 0. If it were true that F = curl G, then this theorem would give div F = div curl G = 0 which contradicts div F = v0. Therefore F is not the curl of another vector field.
Transcribed Image Text:EXAMPLE 5 Show that the vector field F(x, y, z) = xz'i + x'yz°j - yk can't be written as the curl of another vector field, that is, F * curl G. SOLUTION As we showed in this example, it can be shown that div F = and therefore div F = v 0. If it were true that F = curl G, then this theorem would give div F = div curl G = 0 which contradicts div F = v0. Therefore F is not the curl of another vector field.
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