Calculus with Analytic Geometry by edwards & Penney soln ch15

Calculus with Analytic Geometry

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-6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 Section 15.1 C15S01.001: F ( x, y ) = 1 , 1 is a constant vector field; some vectors in this field are shown next. C15S01.002: The vector field F ( x, y ) = 3 , 2 is a constant vector field. Some typical vectors in this field are shown next. 1
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-6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 C15S01.003: Some typical vectors in the field F ( x, y ) = x, y are shown next. C15S01.004: Some typical vectors in the field F ( x, y ) = 2 , x are shown next. C15S01.005: 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 2
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-6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 that each vector points directly away from the origin. C15S01.006: 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. C15S01.007: The vector field F ( x, y, z ) = 0 , 1 , 1 is a constant vector field. All vectors in this field are 3
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x y z (0,1,1) x y z (1,1,0) (0,0,1) parallel translates of the one shown in the next figure. C15S01.008: 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. C15S01.009: 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. C15S01.010: 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. C15S01.011: The vector field ( xy ) = y, x is shown in Fig. 15.1.8. To verify this, evaluate the gradient at (2 , 0). C15S01.012: The gradient vector field (2 x 2 + y 2 ) = 4 x, 2 y is shown in Fig. 15.1.9. To verify this, evaluate the gradient at (2 , 2). C15S01.013: 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 ) is shown in Fig. 15.1.10. To verify this, evaluate the gradient at (1 , 1) and at (0 , 1). 4
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C15S01.014: 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. 15.1.7. To verify this, evaluate the gradient at the point (1 , 1). C15S01.015: 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 . C15S01.016: 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 . C15S01.017: 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 . C15S01.018: 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 . C15S01.019: 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 .
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