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 )= h 1 , 1 i is a constant vector feld; some vectors in this feld are shown next. C15S01.002: The vector feld F ( x, y h 3 , 2 i is a constant vector feld. Some typical vectors in this feld are shown next. 1
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-6 6 -6 -4 -2 2 4 6 -6 2 -6 C15S01.003: Some typical vectors in the feld F ( x, y )= h x, y i are shown next. C15S01.004: Some typical vectors in the feld F ( x, y h 2 ,x i are shown next. C15S01.005: 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 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 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. C15S01.007: The vector Feld F ( x, y, z )= h 0 , 1 , 1 i is a constant vector Feld. All vectors in this Feld 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 Fgure. C15S01.008: 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. C15S01.009: 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. C15S01.010: 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. C15S01.011: The vector Feld ( xy h y, x i is shown in ±ig. 15.1.8. To verify this, evaluate the gradient at (2 , 0). C15S01.012: The gradient vector Feld (2 x 2 + y 2 h 4 x, 2 y i is shown in ±ig. 15.1.9. To verify this, evaluate the gradient at (2 , 2). C15S01.013: 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 ) ® is shown in ±ig. 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 feld ( 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 veri±y this, evaluate the gradient at the point (1 , 1). C15S01.015: F ( x, y, z )= h x, y, z i , then ∇· F =1+1+1=3 and ∇× F = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ijk ∂x ∂y ∂z xyz ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = h 0 , 0 , 0 i = 0 . C15S01.016: F ( x, y, z h 3 x, 2 y, 4 z i , then ∇· F =3 2 4= 3 and ∇× F = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 3 x 2 y 4 z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = h 0 , 0 , 0 i = 0 . C15S01.017: F ( x, y, z h yz, xz, xy i , then ∇· F =0+0+0=0 and ∇× F = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ yz xz xy ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = h x x, y y, z z i = 0 . C15S01.018: F ( x, y, z h x 2 2 ,z 2 i , then ∇· F =2 x +2 y z and ∇× F = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ x 2 y 2 z 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = h 0 , 0 , 0 i = 0 .
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Calculus with Analytic Geometry by edwards & Penney soln ch15

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