57327-0136147054_14

57327-0136147054_14 - Section 14.1 14.1.1: F(x, y ) = 1, 1...

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Section 14.1 14.1.1: F ( x, y )= h 1 , 1 i is a constant vector feld; some vectors in this feld are shown next. -6 -4 -2 2 4 6 x -6 -4 -2 2 4 6 y 14.1.2: The vector feld F ( x, y h 3 , 2 i is a constant vector feld. Some typical vectors in this feld are shown next. -6 -4 -2 2 4 6 8 x -6 -4 -2 2 4 6 y 14.1.3: Some typical vectors in the feld F ( x, y h x, y i are shown next. 1828
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-10 -6 -2 2 6 10 x -6 -2 -4 2 4 6 y 14.1.4: Some typical vectors in the feld F ( x, y )= h 2 ,x i are shown next. -4 -2 2 4 6 x -6 -3 3 6 y 14.1.5: Some typical vectors in the feld F ( x, y h ( x 2 + y 2 ) 1 / 2 h x, y i are shown next. Note that the length oF each vector is proportional to the square oF the distance From the origin to its initial point and that each vector points directly away From the origin. 1829
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-8 -4 4 8 x -8 -4 4 8 y 14.1.6: Some typical vectors in the feld F ( x, y )=( x 2 + y 2 ) 1 / 2 h x, y i are shown next. Note that each is a unit vector that points directly away From the origin. -4 -2 2 4 x -4 -2 2 4 y 14.1.7: The vector feld F ( x, y, z )= h 0 , 1 , 1 i is a constant vector feld. All vectors in this feld are parallel translates oF the one shown in the next fgure. 1830
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H 0,1,1 L x y z 14.1.8: The vector feld F ( x, y, z )= h 1 , 1 , 1 i is a constant vector feld. All vectors in this feld are parallel translates oF the one shown in the next fgure. H 1,1,0 L H 0,0,1 L x y z 14.1.9: Each vector in the feld F ( x, y, z h− x, y i is parallel to the xy -plane and reaches From its initial point at ( x, y, z ) to its terminal point (0 , 0 ,z )onthe z -axis. 14.1.10: Each vector in the feld F ( x, y, z h x, y, z i points directly away From the origin and its length is the same as the distance From the origin to its initial point. 14.1.11: The vector feld ( xy h y, x i is shown in ±ig. 14.1.8. To veriFy this, evaluate the gradient at (2 , 0). 14.9.12: The gradient vector feld (2 x 2 + y 2 h 4 x, 2 y i is shown in ±ig. 14.1.9. To veriFy this, evaluate the gradient at (2 , 2). 14.9.13: The gradient vector feld ( sin 1 2 ( x 2 + y 2 ) ) = ± x cos 1 2 ( x 2 + y 2 ) ,y cos 1 2 ( x 2 + y 2 ) ² 1831
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is shown in Fig. 14.1.10. To verify this, evaluate the gradient at (1 , 1) and at (0 , 1). 14.9.14: The gradient vector ±eld ( sin 1 2 ( y 2 x 2 ) ) = ± x cos 1 2 ( y 2 x 2 ) ,y cos 1 2 ( y 2 x 2 ) ² is shown in Fig. 14.1.7. To verify this, evaluate the gradient at the point (1 , 1). 14.9.15: If F ( x, y, z )= h x, y, z i ,th en ∇· F =1+1+1=3 and ∇× F = ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ ijk ∂x ∂y ∂z xyz ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ = h 0 , 0 , 0 i = 0 . 14.1.16: If F ( x, y, z h 3 x, 2 y, 4 z i F =3 2 4= 3a n d F = ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ 3 x 2 y 4 z ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ = h 0 , 0 , 0 i = 0 . 14.1.17: If F ( x, y, z h yz, xz, xy i F =0+0+0=0 F = ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ yz xz xy ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ = h x x, y y, z z i = 0 . 14.1.18: If F ( x, y, z h x 2 2 ,z 2 i F =2 x +2 y z and F = ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ x 2 y 2 z 2 ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ = h 0 , 0 , 0 i = 0 .
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57327-0136147054_14 - Section 14.1 14.1.1: F(x, y ) = 1, 1...

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