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57327-0136147054_14 - Section 14.1 14.1.1 F(x y = 1 1 is a...

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Section 14.1 14.1.1: F ( x, y ) = 1 , 1 is a constant vector field; some vectors in this field are shown next. -6 -4 -2 2 4 6 x -6 -4 -2 2 4 6 y 14.1.2: The vector field F ( x, y ) = 3 , 2 is a constant vector field. Some typical vectors in this field 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 field F ( x, y ) = x, y 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 field F ( x, y ) = 2 , x are shown next. -4 -2 2 4 6 x -6 -3 3 6 y 14.1.5: Some typical vectors in the field F ( x, y ) = ( x 2 + y 2 ) 1 / 2 x, y 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 field F ( x, y ) = ( x 2 + y 2 ) 1 / 2 x, y 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 field F ( x, y, z ) = 0 , 1 , 1 is a constant vector field. All vectors in this field are parallel translates of the one shown in the next figure. 1830
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0,1,1 x y z 14.1.8: The vector field F ( x, y, z ) = 1 , 1 , 1 is a constant vector field. All vectors in this field are parallel translates of the one shown in the next figure. 1,1,0 0,0,1 x y z 14.1.9: Each vector in the field F ( x, y, z ) = x, y is parallel to the xy -plane and reaches from its initial point at ( x, y, z ) to its terminal point (0 , 0 , z ) on the z -axis. 14.1.10: Each vector in the field F ( x, y, z ) = x, y, z 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 field ( xy ) = y, x is shown in Fig. 14.1.8. To verify this, evaluate the gradient at (2 , 0). 14.9.12: The gradient vector field (2 x 2 + y 2 ) = 4 x, 2 y is shown in Fig. 14.1.9. To verify this, evaluate the gradient at (2 , 2). 14.9.13: The gradient vector field ( sin 1 2 ( x 2 + y 2 ) ) = x cos 1 2 ( x 2 + y 2 ) , y cos 1 2 ( x 2 + y 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 field ( sin 1 2 ( y 2 x 2 ) ) = x cos 1 2 ( y 2 x 2 ) , y cos 1 2 ( y 2 x 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 ) = x, y, z , then ∇· F = 1 + 1 + 1 = 3 and ∇× F = i j k ∂x ∂y ∂z x y z = 0 , 0 , 0 = 0 . 14.1.16: If F ( x, y, z ) = 3 x, 2 y, 4 z , then ∇· F = 3 2 4 = 3 and ∇× F = i j k ∂x ∂y ∂z 3 x 2 y 4 z = 0 , 0 , 0 = 0 . 14.1.17: If F ( x, y, z ) = yz, xz, xy , then ∇· F = 0 + 0 + 0 = 0 and ∇× F = i j k ∂x ∂y ∂z yz xz xy = x x, y y, z z = 0 . 14.1.18: If F ( x, y, z ) = x 2 , y 2 , z 2 , then ∇· F = 2 x + 2 y + 2 z and ∇× F = i j k ∂x ∂y ∂z x 2 y 2 z 2 = 0 , 0 , 0 = 0 . 14.1.19: If F ( x, y, z ) = xy 2 , yz 2 , zx 2 , then ∇· F = y 2 + z 2 + x 2 and ∇× F = i j k ∂x ∂y ∂z xy 2 yz 2 zx 2 = 2 yz, 2 xz, 2 xy . 1832
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14.1.20: If F ( x, y, z ) = 2 x y, 3 y 2 z, 7 z 3 x , then ∇· F = 2 + 3 + 7 = 12 and ∇× F = i j k ∂x ∂y ∂z 2 x y 3 y 2 z 7 z 3 x = 2 , 3 , 1 .
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