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conservative,

conservative,

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Find a function f such that F = Vf. F(x, y, z) = 4y2z³i + 8xyz³j + 12xy2z2k Step 1 Since all the component functions of F have continuous partials, then F will be conservative if curl(F) = 0 Step 2 For F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k = 4y2z³i + 8xyz³j + 12xy2z?k, we have the following. ƏR ay dz aR dz дх aQ ap ax ay Step 3 Since all components are 0, we conclude that curl(F) = 0 and, therefore, F is conservative. Thus, a potential function f(x, y, z) exists for which fo(x, y, z) = 4vz This means that f(x, y, z) = 4v z°x + g(y, z). Step 4 We have - 4xy²z3 + g(y, 2) = 8xyz · gyly, z). Also, since F is conservative, we know that fy(x, y, z) = 8xyz Therefore, gyly, z) = 0 Step 5 Since g,(y, z) = 0, then it must be true that g(y, z) = h(z). This means that f(x, y, z) = 4xy²z3 + h(z), and so f,(x, y, z) = 12xy z? Step 6 Since F is conservative, we know that f,(x, y, z) = 12yz? Therefore, h'(z) = 0
Transcribed Image Text:Find a function f such that F = Vf. F(x, y, z) = 4y2z³i + 8xyz³j + 12xy2z2k Step 1 Since all the component functions of F have continuous partials, then F will be conservative if curl(F) = 0 Step 2 For F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k = 4y2z³i + 8xyz³j + 12xy2z?k, we have the following. ƏR ay dz aR dz дх aQ ap ax ay Step 3 Since all components are 0, we conclude that curl(F) = 0 and, therefore, F is conservative. Thus, a potential function f(x, y, z) exists for which fo(x, y, z) = 4vz This means that f(x, y, z) = 4v z°x + g(y, z). Step 4 We have - 4xy²z3 + g(y, 2) = 8xyz · gyly, z). Also, since F is conservative, we know that fy(x, y, z) = 8xyz Therefore, gyly, z) = 0 Step 5 Since g,(y, z) = 0, then it must be true that g(y, z) = h(z). This means that f(x, y, z) = 4xy²z3 + h(z), and so f,(x, y, z) = 12xy z? Step 6 Since F is conservative, we know that f,(x, y, z) = 12yz? Therefore, h'(z) = 0
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