# Chapter 15 - SECTION 15.1 (PAGE 811) R. A. ADAMS: CALCULUS...

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SECTION 15.1 (PAGE 811) R. A. ADAMS: CALCULUS CHAPTER 15. VECTOR FIELDS Section 15.1 Vector and Scalar Fields (page 811) 1. F = x i + x j . The ﬁeld lines satisfy dx x = dy x , i.e., = . The ﬁeld lines are y = x + C , straight lines parallel to y = x . y x Fig. 15.1.1 2. F = x i + y j . The ﬁeld lines satisfy x = y . Thus ln y = ln x + ln C ,o r y = Cx . The ﬁeld lines are straight half-lines emanating from the origin. y x Fig. 15.1.2 3. F = y i + x j . The ﬁeld lines satisfy y = x . Thus xdx = ydy . The ﬁeld lines are the rectangular hyperbolas (and their asymptotes) given by x 2 y 2 = C . y x Fig. 15.1.3 4. F = i + sin x j . The ﬁeld lines satisfy = sin x . Thus = sin x . The ﬁeld lines are the curves y =− cos x + C . y x Fig. 15.1.4 5. F = e x i + e x j . The ﬁeld lines satisfy e x = e x . Thus = e 2 x . The ﬁeld lines are the curves y 1 2 e 2 x + C . y x Fig. 15.1.5 6. F =∇ ( x 2 y ) = 2 x i j . The ﬁeld lines satisfy 2 x = 1 . They are the curves y 1 2 ln x + C . y x Fig. 15.1.6 570

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INSTRUCTOR’S SOLUTIONS MANUAL SECTION 15.1 (PAGE 811) 7. F =∇ ln ( x 2 + y 2 ) = 2 x i + 2 y j x 2 + y 2 . The ﬁeld lines satisfy dx x = dy y . Thus they are radial lines y = Cx (and x = 0) y x Fig. 15.1.7 8. F = cos y i cos x j . The ﬁeld lines satisfy cos y =− cos x , that is, cos xdx + cos ydy = 0. Thus they are the curves sin x + sin y = C . y x Fig. 15.1.8 9. v ( x , y , z ) = y i y j y k . The streamlines satisfy dz . Thus y + x = C 1 , z + x = C 2 . The streamlines are straight lines parallel to i j k . 10. v ( x , y , z ) = x i + y j x k . The streamlines satisfy x = y x . Thus z + x = C 1 , y = C 2 x . The streamlines are straight half- lines emanating from the z -axis and perpendicular to the vector i + k . 11. v ( x , y , z ) = y i x j + k . The streamlines satisfy y x = . Thus + = 0, so x 2 + y 2 = C 2 1 . Therefore, = 1 y = 1 ± C 2 1 x 2 . This implies that z = sin 1 x C 1 + C 2 . The streamlines are the spirals in which the surfaces x = C 1 sin ( z C 2 ) intersect the cylinders x 2 + y 2 = C 2 1 . 12. v = x i + y j ( 1 + z 2 )( x 2 + y 2 ) . The streamlines satisfy = 0 and x = y . Thus z = C 1 and y = C 2 x . The streamlines are horizontal half-lines emanating from the z -axis. 13. v = xz i + yz j + x k . The ﬁeld lines satisfy = = x , or, equivalently, / x = / y and = zdz . Thus the ﬁeld lines have equations y = C 1 x ,2 x = z 2 + C 2 , and are therefore parabolas. 14. v = e xyz ( x i + y 2 j + z k ) . The ﬁeld lines satisfy x = y 2 = z , so they are given by z = C 1 x ,ln | x |= ln | C 2 |− ( 1 / y ) (or, equivalently, x = C 2 e 1 / y ). 15. v ( x , y ) = x 2 i y j . The ﬁeld lines sat- isfy / x 2 / y , so they are given by ln | y ( 1 / x ) + ln | C | ,o r y = Ce 1 / x . 16. v ( x , y ) = x i + ( x + y ) j . The ﬁeld lines satisfy x = x + y = x + y x Let y = x v( x ) = v + x d v v + x d v = x ( 1 + v) x = 1 + v. Thus d v/ = 1 / x , and so x ) = ln | x |+ C . The ﬁeld lines have equations y = x ln | x . 17. F = ˆ r + r ˆ θ . The ﬁeld lines satisfy dr = d θ , so they are the spirals r = θ + C . 18. F = ˆ r + θ ˆ θ . The ﬁeld lines satisfy = rd θ/θ r / r = d , so they are the spirals r = C θ . 19. F = 2 ˆ r + θ ˆ θ . The ﬁeld lines satisfy / 2 = r / r = 2 d , so they are the spirals r = C θ 2 .
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## This note was uploaded on 12/16/2009 for the course FEW, FEWEB 400567 taught by Professor Moerdersen during the Fall '09 term at Vrije Universiteit Amsterdam.

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Chapter 15 - SECTION 15.1 (PAGE 811) R. A. ADAMS: CALCULUS...

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