Irrotational field and scalar potential F=(x²−yz)i+(y²−zx)j…

A vector field F is irrotational if ∇×F = 0. Then F = ∇φ for some scalar potential φ.

F = (x² − yz) i + (y² − zx) j + (z² − xy) k

Curl

∇×F = | i          j          k        |
      | ∂/∂x      ∂/∂y      ∂/∂z      |
      | x²−yz     y²−zx     z²−xy     |

i : ∂(z²−xy)/∂y − ∂(y²−zx)/∂z = −x − (−x) = 0
j : ∂(x²−yz)/∂z − ∂(z²−xy)/∂x = −y − (−y) = 0
k : ∂(y²−zx)/∂x − ∂(x²−yz)/∂y = −z − (−z) = 0

So ∇×F = 0 and F is irrotational.

Scalar potential φ with F = ∇φ

∂φ/∂x = x² − yz   ... (1)
∂φ/∂y = y² − zx   ... (2)
∂φ/∂z = z² − xy   ... (3)

Integrate (1) with respect to x: φ = x³/3 − xyz + f(y, z).

Differentiate with respect to y and compare with (2): −xz + ∂f/∂y = y² − zx ⇒ ∂f/∂y = y² ⇒ f = y³/3 + g(z).

So φ = x³/3 + y³/3 − xyz + g(z). Differentiate with respect to z and compare with (3): −xy + g′(z) = z² − xy ⇒ g′(z) = z² ⇒ g = z³/3 + c.

Scalar potential: φ = (x³ + y³ + z³)/3 − xyz + c, where c is an arbitrary constant.

(Check: ∂φ/∂x = x² − yz ✓, ∂φ/∂y = y² − xz ✓, ∂φ/∂z = z² − xy ✓.)