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EXAMPLE 2 Finding potential functions Find a potential function for the conserva- tive vector fields in Example 1. a. F(ecos y, e* sin y) - b. F (2xy z2, x² + 2z, 2y2rz) SOLUTION a. A potential function op for F = conditions (f.g) has the property that F = Vp and satisfies the = f(x, y) = e cosy and y = g(x, y) = -e" sin y. The first equation is integrated with respect to x (holding y fixed) to obtain which implies that So, dx = Sex cos y dx, = (x, y) e cos y + c(y). In this case, the "constant of integration" c(y) is an arbitrary function of; can check the preceding calculation by noting that ар ax == a - (e' cos y + c(y)) e cos y = f(x, y). y. You To find the arbitrary function c(y), we differentiate (x, y) ecos y c(y) with respect to y and equate the result to g (recall that y = g): yesin yc'(y) and gesin y. We conclude that c'(y) = 0, which implies that c(y) is any real number, which we typically take to be zero. So a potential function is p(x, y) = cos y, a result that may be checked by differentiation.

EXAMPLE 2 Finding potential functions Find a potential function for the conserva- tive vector fields in Example 1. a. F(ecos y, e* sin y) - b. F (2xy z2, x² + 2z, 2y2rz) SOLUTION a. A potential function op for F = conditions (f.g) has the property that F = Vp and satisfies the = f(x, y) = e cosy and y = g(x, y) = -e" sin y. The first equation is integrated with respect to x (holding y fixed) to obtain which implies that So, dx = Sex cos y dx, = (x, y) e cos y + c(y). In this case, the "constant of integration" c(y) is an arbitrary function of; can check the preceding calculation by noting that ар ax == a - (e' cos y + c(y)) e cos y = f(x, y). y. You To find the arbitrary function c(y), we differentiate (x, y) ecos y c(y) with respect to y and equate the result to g (recall that y = g): yesin yc'(y) and gesin y. We conclude that c'(y) = 0, which implies that c(y) is any real number, which we typically take to be zero. So a potential function is p(x, y) = cos y, a result that may be checked by differentiation.

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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Calculus III

May I please have some elaborations on Example 2 part a? 
Thank you.

EXAMPLE 2 Finding potential functions Find a potential function for the conserva- tive vector fields in Example 1. a. F(ecos y, e* sin y) - b. F (2xy z2, x² + 2z, 2y2rz) SOLUTION a. A potential function op for F = conditions (f.g) has the property that F = Vp and satisfies the = f(x, y) = e cosy and y = g(x, y) = -e" sin y. The first equation is integrated with respect to x (holding y fixed) to obtain which implies that So, dx = Sex cos y dx, = (x, y) e cos y + c(y). In this case, the "constant of integration" c(y) is an arbitrary function of; can check the preceding calculation by noting that ар ax == a - (e' cos y + c(y)) e cos y = f(x, y). y. You To find the arbitrary function c(y), we differentiate (x, y) ecos y c(y) with respect to y and equate the result to g (recall that y = g): yesin yc'(y) and gesin y. We conclude that c'(y) = 0, which implies that c(y) is any real number, which we typically take to be zero. So a potential function is p(x, y) = cos y, a result that may be checked by differentiation.
Transcribed Image Text:EXAMPLE 2 Finding potential functions Find a potential function for the conserva- tive vector fields in Example 1. a. F(ecos y, e* sin y) - b. F (2xy z2, x² + 2z, 2y2rz) SOLUTION a. A potential function op for F = conditions (f.g) has the property that F = Vp and satisfies the = f(x, y) = e cosy and y = g(x, y) = -e" sin y. The first equation is integrated with respect to x (holding y fixed) to obtain which implies that So, dx = Sex cos y dx, = (x, y) e cos y + c(y). In this case, the "constant of integration" c(y) is an arbitrary function of; can check the preceding calculation by noting that ар ax == a - (e' cos y + c(y)) e cos y = f(x, y). y. You To find the arbitrary function c(y), we differentiate (x, y) ecos y c(y) with respect to y and equate the result to g (recall that y = g): yesin yc'(y) and gesin y. We conclude that c'(y) = 0, which implies that c(y) is any real number, which we typically take to be zero. So a potential function is p(x, y) = cos y, a result that may be checked by differentiation.
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