Download Question (a) Find the constants a,b,c so that the force feild defined

Question
(a) Find the constants a,b,c so that the force feild defined by
(x + 2y + az)i + (bx − 3y − z)j + (4x + cy + 2z)k
is conservative.
(b) What is the potential associated with this force field when a, b, c are
chosen to make it conservative.
Answer
We require ∇ × F = 0
0 =
¯
¯
i
¯
¯
∂
¯
∂x
¯
¯ (x + 2y + az)
j
k
∂
∂y
(bx − 3y − z)
= i(c + 1) + j(a − 4) + k(b − 2)
¯
¯
¯
¯
∂
¯
∂z
¯
(4x + cy + 2z) ¯
Therefore a = 4, b = 2, c = −1.
F = (x + 2y + 4z)i + (2x − 3y − z)j + (4x − 1y + 2z)k
∂U
∂U
∂U
i−
j−
k
= −
∂x
∂y
∂z
∂u
= −(x + 2y + 4z)
∂x
⇒ U = − 12 x2 − 2xy − 4xz + f1 (y, z) (1)
−
∂U
= −(2x − 3y − 2)
∂y
⇒ U = −2xy + 32 y 2 + yz + f2 (x, z)
−
(2)
∂U
= −(4x − y + 2z)
∂z
⇒ U = −4xz + yz + 21 z 2 + f3 (x, z) (3)
Comparing:
(1) and (2) ⇒ f1 (y, z) = 23 y 2 + yz + g(z)
f2 (x, z) = −4xz − 21 x2 g(z)
(2) and (3) ⇒ f3 (x, y) = −2xy + 32 y 2 − 12 x2 , g(z) = − 12 z 2
Therefore U = −4x2 + y 2 − 2xy − 12 x2 + 23 y 2 − 21 z 2
−
1