6-70. Consider the tetrahedron defined by the vectors A, B, C illustrated.
(a) Show the vectors n 1 =^12 A×B, n 2 =^12 B×C ,
n 3 =^12 C×A, n 4 =^12 (C−A)×(B−A)are normal to the faces
of the tetrahedron with magnitudes equal to the area of the
faces. (b) Show n 1 +n 2 +n 3 +n 4 = 0
6-71. Find the work done in moving a particle in a counterclockwise direction
around a unit circle in the z= 0 plane if the particle moves in the force field
F=F(x, y, z ) = (x+y+z)ˆe 1 + (2 x−y+ 3z)ˆe 2 + (3 x−y−z)ˆe 3.
6-72. The straight-line defined by the parametric equations
x= 2 + λ, y = 3 + 2 λ, z = 4 − 2 λ
with parameter λ, is drawn through the force field F =F(x, y, z ) = xy ˆe 1 +yz ˆe 2 +zˆe 3.
Evaluate the given line integrals along this line from the point P 1 (2 , 3 ,4) to the point
P 2 (4 , 7 ,0)
(a)
∫P 2
P 1
(x^2 +y^2 )ds (b)
∫P 2
P 1
F·dr (c)
∫P 2
P 1
F×dr
6-73. A particle moves around the closed
curve Cillustrated in figure. It moves in a vector
field F defined by
F =F(x, y ) = 6(y^2 −x)ˆe 1 + 6 xˆe 2.
Evaluate the line integrals in parts (a) and (b).
(a) ........
...................................
....................
∫
............................ F·dr (b) ...............................................................
∫
............................ F×dr (c) Show that ...............................................................
∫
............................ F·dr =−...............................................................
∫
.............................. F·dr
6-74. Evaluate the given line integrals along the path
C={(x, y )|x= 2t, y = 1 + t+t^2 }from t= 0 to t= 3.
(a)
∫
C
y dx + (x+y)dy (b)
∫
C
y dx −x dy (c)
∫
C
2 xy dx +x^2 dy
6-75. Evaluate the given line integrals around the square with vertices (0 ,0),(1 ,0),
(1 ,1) and (0 ,1), both clockwise and counterclockwise.
(a)
∫
C
©x(y+ 1) dx + (x+ 1)y dy (b)
∫
C
©(x^2 −y^2 )dx + (x^2 +y^2 )dy (c)
∫
C
©y dx +x dy
