chapter_16 - January 27, 2005 11:56 L24-CH16 Sheet number 1...

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January 27, 2005 11:56 L24-CH16 Sheet number 1 Page number 693 black 693 CHAPTER 16 Topics in Vector Calculus EXERCISE SET 16.1 1. (a) III because the vector feld is independent oF y and the direction is that oF the negative x -axis For negative x , and positive For positive (b) IV, because the y -component is constant, and the x -component varies periodically with x 2. (a) I, since the vector feld is constant (b) II, since the vector feld points away From the origin 3. (a) true (b) true (c) true 4. (a) False, the lengths are equal to 1 (b) False, the y -component is then zero (c) False, the x -component is then zero 5. x y 6. x y 7. x y 8. x y 9. x y 10. x y 11. (a) φ = φ x i + φ y j = y 1+ x 2 y 2 i + x x 2 y 2 j = F ,so F is conservative For all x,y (b) φ = φ x i + φ y j =2 x i 6 y j +8 z k = F so F is conservative For all 12. (a) φ = φ x i + φ y j =(6 xy y 3 ) i +(4 y +3 x 2 3 xy 2 ) j = F F is conservative For all (b) φ = φ x i + φ y j + φ z k = (sin z + y cos x ) i + (sin x + z cos y ) j +( x cos z + sin y ) k = F F is conservative For all 13. div F x + y , curl F = z i 14. div F = z 3 y 3 x 2 +10 zy , curl F =5 z 2 i xz 2 j +4 xy 4 k 15. div F = 0, curl F = (40 x 2 z 4 12 xy 3 ) i + (14 y 3 z y 4 ) j (16 xz 5 +21 y 2 z 2 ) k 16. div F = ye xy + sin y + 2 sin z cos z , curl F = xe xy k
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January 27, 2005 11:56 L24-CH16 Sheet number 2 Page number 694 black 694 Chapter 16 17. div F = 2 p x 2 + y 2 + z 2 , curl F = 0 18. div F = 1 x + xze xyz + x x 2 + z 2 , curl F = xye xyz i + z x 2 + z 2 j + yze xyz k 19. · ( F × G )= · ( ( z +4 y 2 ) i +(4 xy +2 xz ) j +(2 xy x ) k )=4 x 20. · ( F × G · (( x 2 yz 2 x 2 y 2 ) i xy 2 z 2 j + xy 2 z k xy 2 21. · ( ∇× F · ( sin( x y ) k )=0 22. · ( F · ( ze yz i + xe xz j +3 e y k 23. ( F ( xz i j + y k )=(1+ y ) i + x j 24. ( F (( x y ) i y j 2 xy k 2 x i y j 3 k 27. Let F = f i + g j + h k ; div ( k F k ∂f ∂x + k ∂g ∂y + k ∂h ∂z = k div F 28. Let F = f i + g j + h k ; curl ( k F k µ i + k µ j + k µ k = k curl F 29. Let F = f ( x,y,z ) i + g ( ) j + h ( ) k and G = P ( ) i + Q ( ) j + R ( ) k , then div ( F + G µ + ∂P + µ + ∂Q + µ + ∂R = µ + + + µ + + = div F + div G 30. Let F = f ( ) i + g ( ) j + h ( ) k and G = P ( ) i + Q ( ) j + R ( ) k , then curl ( F + G · ( h + R ) ( g + Q ) ¸ i + · ( f + P ) ( h + R ) ¸ j + · ( g + Q ) ( f + P ) ¸ k ; expand and rearrange terms to get curl F + curl G . 31. Let F = f i + g j + h k ; div ( φ F µ φ + ∂φ f + µ φ + g + µ φ + h = φ µ + + + µ f + g + h = φ div F + φ · F 32. Let F = f i + g j + h k ; curl ( φ F · ( φh ) ( φg ) ¸ i + · ( φf ) ( φh ) ¸ j + · ( φg ) ( φf ) ¸ k ; use the product rule to expand each of the partial derivatives, rearrange to get φ curl F + φ × F
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January 27, 2005 11:56 L24-CH16 Sheet number 3 Page number 695 black Exercise Set 16.1 695 33. Let F = f i + g j + h k ; div(curl F )= ∂x µ ∂h ∂y ∂g ∂z + µ ∂f + µ = 2 h ∂x∂y 2 g ∂x∂z + 2 f ∂y∂z 2 h ∂y∂x + 2 g ∂z∂x 2 f ∂z∂y =0 , assuming equality of mixed second partial derivatives 34. curl ( φ µ 2 φ 2 φ i + µ 2 φ 2 φ j + µ 2 φ 2 φ k = 0 , assuming equality of mixed second partial derivatives 35. · ( k F k · F , · ( F + G · F + · G ,
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chapter_16 - January 27, 2005 11:56 L24-CH16 Sheet number 1...

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