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a math/eng question im...

a math/eng question im not sure about.

(a)Evaluatethe scalar line integral of the vector field

F1

(x, y, z)= 2y i +3x j +4z k

around the closed circular path C of radius1 centred at the point

(1, 0, 2) defined byt he parametric equations

x =1 +cos t, y =sin t, z =2 for −π ≤ t ≤ π.

What can youdeduceabout thevector field F1

?[6 ]

(b)Consider vector field

F2

(x, y, z)= (2xy

2

− 3yz +1) i +(2yx

2

− 3xz) j − 3xy k.

(i)Calculate curl F2

,and statewhether or not F2

is conserv ative. [5 ]

(ii)Calculatethe scalar line integral of F2

alongthe path consisting

of thestraight-linesegmentjoiningthe origin to thepoint(a, b, c). [4 ]

(iii) Hence find apotential function U (x, y, z)suchthat

F2 = − grad U .Using the expression for U (x, y, z)thatyou

found, ev aluate − grad U andconfirmthat F2 = − grad U .[3 ]

(c) Avector field F3

is expressedincylindrical polar coordinates as

F3

(ρ, θ, z)= ρ

2

(cos(4θ)+ 1) e

ρ

+2ρ sin(4θ) e

θ + z

2

e

z

.

Determine div F3

.[3 ]

(d)A velocity field v is expressedinspherical polar coordinates as

v(r, θ, φ)= 3r

2

sin θ cos φ e

r + r

2

cos θ cos φ e

θ − r

2

sin φ e

φ

.

By calculating curl v,showthat v is everywhere conserv ative. [4 ]

Remember that TMA07containsrevision questionslikethose in

(a)Evaluatethe scalar line integral of the vector field

F1

(x, y, z)= 2y i +3x j +4z k

around the closed circular path C of radius1 centred at the point

(1, 0, 2) defined byt he parametric equations

x =1 +cos t, y =sin t, z =2 for −π ≤ t ≤ π.

What can youdeduceabout thevector field F1

?[6 ]

(b)Consider vector field

F2

(x, y, z)= (2xy

2

− 3yz +1) i +(2yx

2

− 3xz) j − 3xy k.

(i)Calculate curl F2

,and statewhether or not F2

is conserv ative. [5 ]

(ii)Calculatethe scalar line integral of F2

alongthe path consisting

of thestraight-linesegmentjoiningthe origin to thepoint(a, b, c). [4 ]

(iii) Hence find apotential function U (x, y, z)suchthat

F2 = − grad U .Using the expression for U (x, y, z)thatyou

found, ev aluate − grad U andconfirmthat F2 = − grad U .[3 ]

(c) Avector field F3

is expressedincylindrical polar coordinates as

F3

(ρ, θ, z)= ρ

2

(cos(4θ)+ 1) e

ρ

+2ρ sin(4θ) e

θ + z

2

e

z

.

Determine div F3

.[3 ]

(d)A velocity field v is expressedinspherical polar coordinates as

v(r, θ, φ)= 3r

2

sin θ cos φ e

r + r

2

cos θ cos φ e

θ − r

2

sin φ e

φ

.

By calculating curl v,showthat v is everywhere conserv ative. [4 ]

Remember that TMA07containsrevision questionslikethose in

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