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Line integral independent of path

Short answer

Curl F = 0, so the integral is path-independent. A potential function is φ = x³yz², and the line integral from (0,0,0) to (1,2,3) equals φ(1,2,3) − φ(0,0,0) = 18.

If F = 3x²yz² i + x³z² j + 2x³yz k, show that ∫_C F·dr is independent of the path of integration. Hence evaluate the integral on any path C from (0,0,0) to (1,2,3).

A line integral ∫ F·dr is independent of path if F is conservative, that is curl F = 0 (then F = ∇φ).

Test

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 the integral is independent of the path.

Potential function

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

Value

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

Answer: the integral is path independent and equals 18.

✓ Answer checked by Claude Opus 5.5 (high effort)

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