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Irrotational field and scalar potential F=(2xz³+6y)i+(6x−2yz)j…

Short answer

Curl F = 0, so it is conservative. Integrate ∂φ/∂x = 2xz³+6y → φ = x²z³+6xy+f(y,z); ∂φ/∂y = 6x−2yz → f = −y²z+g(z); ∂φ/∂z = 3x²z²−y² matches, so g'=0. Potential: φ = x²z³+6xy−y²z+c.

Prove that F = (2xz³ + 6y) i + (6x − 2yz) j + (3x²z² − y²) k is a conservative vector field and find its scalar potential function.

A vector field is conservative if curl F = 0 (the region is simply connected), and then F = ∇φ.

P = 2xz³ + 6y, Q = 6x − 2yz, R = 3x²z² − y².

i: ∂R/∂y − ∂Q/∂z = −2y − (−2y) = 0
j: ∂P/∂z − ∂R/∂x = 6xz² − 6xz² = 0
k: ∂Q/∂x − ∂P/∂y = 6 − 6 = 0

curl F = 0, so F is conservative.

Scalar potential

∂φ/∂x = 2xz³ + 6y   → φ = x²z³ + 6xy + f(y,z)
∂φ/∂y = 6x + f_y = 6x − 2yz  → f_y = −2yz → f = −y²z + g(z)
∂φ/∂z = 3x²z² − y² + g'(z) = 3x²z² − y²  → g' = 0

Answer: φ = x²z³ + 6xy − y²z + c. Check: ∇φ = (2xz³ + 6y) i + (6x − 2yz) j + (3x²z² − y²) k = F.

✓ Answer checked by Claude Opus 5.5 (high effort)

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