chap16-et-student-solutions

chap16-et-student-solutions - LINE AND 17 SURFACE INTEGRALS...

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17 LINE AND SURFACE INTEGRALS 17.1 Vector Fields (ET Section 16.1) Preliminary Questions 1. Which of the following is a unit vector Feld in the plane? (a) F = h y , x i (b) F = * y p x 2 + y 2 , x p x 2 + y 2 + (c) F = ¿ y x 2 + y 2 , x x 2 + y 2 À SOLUTION (a) The length of the vector h y , x i is kh y , x ik = q y 2 + x 2 This value is not 1 for all points, hence it is not a unit vector Feld. (b) We have ° ° ° ° ° * y p x 2 + y 2 , x p x 2 + y 2 + ° ° ° ° ° = v ± ± t à y p x 2 + y 2 ² 2 + à x p x 2 + y 2 ² 2 = s y 2 x 2 + y 2 + x 2 x 2 + y 2 = s y 2 + x 2 x 2 + y 2 = 1 Hence the Feld is a unit vector Feld, for ( x , y ) 6= ( 0 , 0 ) . (c) We compute the length of the vector: ° ° ° ° ¿ y x 2 + y 2 , x x 2 + y 2 À ° ° ° ° = s µ y x 2 + y 2 2 + µ x x 2 + y 2 2 = v ± ± t y 2 + x 2 ( x 2 + y 2 ) 2 = s 1 x 2 + y 2 Since the length is not identically 1, the Feld is not a unit vector Feld. 2. Sketch an example of a nonconstant vector Feld in the plane in which each vector is parallel to h 1 , 1 i . The non-constant vector h x , x i is parallel to the vector h 1 , 1 i . y x 3. Show that the vector Feld F = h− z , 0 , x i is orthogonal to the position vector −→ OP at each point P .Giveanexample of another vector Feld with this property. The position vector at P = ( x , y , z ) is h x , y , z i . We must show that the following dot product is zero: h x , y , z i · h− z , 0 , x i = x · ( z ) + y · 0 + z · x = 0
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SECTION 17.1 Vector Fields (ET Section 16.1) 563 Therefore, the vector Feld F = h− z , 0 , x i is orthogonal to the position vector. Another vector Feld with this property is F = h 0 , z , y i ,since h 0 , z , y i · h x , y , z i = 0 · x + ( z ) · y + y · z = 0 4. Give an example of a potential function for h yz , xz , xy i other than ϕ ( x , y , z ) = xyz . SOLUTION Since any two potential functions of a gradient vector Feld differ by a constant, a potential function for the given Feld other than φ ( x , y , z ) = is, for instance, 1 ( x , y , z ) = + 1. Exercises 1. Compute and sketch the vector assigned to the points P = ( 1 , 2 ) and Q = ( 1 , 1 ) by the vector Feld F = - x 2 , x ® . The vector assigned to P = ( 1 , 2 ) is obtained by substituting x = 1in F ,thatis, F ( 1 , 2 ) =h 1 2 , 1 i= h 1 , 1 i Similarly, F ( 1 , 1 ) = - ( 1 ) 2 , 1 ® 1 , 1 i x 1 1 1 y F ( P ) = 1, 1 F ( Q ) = 1, 1 Compute and sketch the vector assigned to the points P = ( 1 , 2 ) and Q = ( 1 , 1 ) by the vector Feld F = h− y , x i . 3. Compute and sketch the vector assigned to the points P = ( 0 , 1 , 1 ) and Q = ( 2 , 1 , 0 ) by the vector Feld F = - , z 2 , x ® . To Fnd the vector assigned to the point P = ( 0 , 1 , 1 ) , we substitute x = 0, y = 1, z = F , z 2 , x i . We get F ( P ) 0 · 1 , 1 2 , 0 i=h 0 , 1 , 0 i Similarly, F ( Q ) is obtained by substituting x = 2, y = 1, z = 0in F .Thatis, F ( Q ) 2 · 1 , 0 2 , 2 2 , 0 , 2 i F ( P ) = 0, 1, 0 F ( Q ) = 2, 0, 2 y x z Compute the vector assigned to the points P = ( 1 , 1 , 0 ) and Q = ( 2 , 1 , 2 ) by the vector Felds e r , e r r ,and e r r 2 . In Exercises 5–13, sketch the following planar vector Felds by drawing the vectors attached to points with integer coor- dinates in the rectangle 3 x , y 3 . Instead of drawing the vectors with their true lengths, scale them if necessary to avoid overlap.
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This note was uploaded on 02/17/2009 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

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chap16-et-student-solutions - LINE AND 17 SURFACE INTEGRALS...

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