Line integral independent of path

A line integral ∫_C F·dr is independent of the path when F is conservative, i.e. curl F = 0.

Part 1: independence (3 marks)

F = 3x²yz² i + x³z² j + 2x³yz k.

i: ∂R/∂y − ∂Q/∂z = 2x³z − 2x³z = 0
j: ∂P/∂z − ∂R/∂x = 6x²yz − 6x²yz = 0
k: ∂Q/∂x − ∂P/∂y = 3x²z² − 3x²z² = 0

curl F = 0, so F = ∇φ for some φ and the integral does not depend on the path.

Part 2: value (2 marks)

∂φ/∂x = 3x²yz²  → φ = x³yz² + f(y,z)
∂φ/∂y = x³z² → f_y = 0;   ∂φ/∂z = 2x³yz → f_z = 0
φ = x³yz²

∫ F·dr from (0,0,0) to (1,2,3) = φ(1,2,3) − φ(0,0,0) = (1)(2)(9) − 0 = 18

Answer: the integral is path independent and equals 18.