Begin2.DVI

9-12. At all points (x, y )between the circles x^2 +y^2 = 1 and x^2 +y^2 = 9, the vector

function F
=

−yˆe 1 +xeˆ 2

x^2 +y^2 is continuous and equals the gradient of the scalar function

Φ(x, y ) = tan−^1

y

x.

Show that

∫(2,0)

(− 2 ,0)

F
·d
r is not independent of the path of integration by computing

this line integral along the upper half and then the lower half of the circle x^2 +y^2 = 4.

Is the region of integration a simply-connected region?

9-13. Find a vector potential for

(a) F
 = 2yˆe 1 + 2 xˆe 2 (b) F
= (x−y)eˆ 1 −zˆe 3

9-14. For the gravity field F
=−mg ˆe 3

((a) Show that this vector field is irrotational.

(b) Find the potential function from which this field is derivable.

(c) Show that the work done in moving from a height h 1 to a height h 2 is the change

in potential energy.

9-15. (Conservation of Energy )

(a) If F
=md

(^2)
r
dt^2

show that F
·d
r

dt

=^1

2

md

dt

(

d
r

dt

) 2

(b) Show

∫(x,y,z)

(x 0 ,y 0 ,z 0 )

F
·d
r =^1

2 m v

2 (x,y,z)

(x 0 ,y 0 ,z 0 )

=

1

2 m v

2

(x,y,z)

−

1

2 m v

2

(x 0 ,y 0 ,z 0 )

(c) If F
is a conservative vector field such that F
=−∇ φ, show that

∫(x,y,z)

(x 0 ,y 0 ,z 0 )

F
·d
r =−

∫(x,y,z)

(x 0 ,y 0 ,z 0 )

∇φ·d
r =−

∫(x,y,z)

(x 0 ,y 0 ,z 0 )

dφ =−φ(x, y, z) + φ(x 0 , y 0 , z 0 )

(d) Show that φ(x 0 , y 0 , z 0 ) +^1

2

m v^2

(x 0 ,y 0 ,z 0 )

=φ(x, y, z ) +^1

2

m v^2

(x,y,z)

which states that

for a conservative vector field the sum of the potential energy and kinetic energy

at point (x 0 , y 0 , z 0 ) is the same as the sum of the potential energy and kinetic

energy at the point (x, y, z ).

9-16. A conservative vector field has the family of equipotential curves

x^2 −y^2 =c.

Find the field lines and vector field associated with this potential.