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Div and curl of grad(x³+y³+z³−3xyz)

Short answer

F = grad(x³+y³+z³−3xyz) = (3x²−3yz, 3y²−3zx, 3z²−3xy). Divergence: div F = 6x+6y+6z = 6(x+y+z). Curl: curl F = 0, since the curl of any gradient is identically zero.

Find the divergence and curl of F where F = grad(x³+y³+z³−3xyz).

Let φ = x³ + y³ + z³ − 3xyz and F = grad φ.

F

∂φ/∂x = 3x² − 3yz
∂φ/∂y = 3y² − 3zx
∂φ/∂z = 3z² − 3xy
F = (3x² − 3yz) i + (3y² − 3zx) j + (3z² − 3xy) k

Divergence

div F = ∂(3x² − 3yz)/∂x + ∂(3y² − 3zx)/∂y + ∂(3z² − 3xy)/∂z
      = 6x + 6y + 6z
      = 6(x + y + z)

Curl

curl F = | i              j              k           |
         | ∂/∂x           ∂/∂y           ∂/∂z        |
         | 3x²−3yz        3y²−3zx        3z²−3xy     |
  • i: ∂(3z² − 3xy)/∂y − ∂(3y² − 3zx)/∂z = −3x + 3x = 0
  • j: ∂(3x² − 3yz)/∂z − ∂(3z² − 3xy)/∂x = −3y + 3y = 0
  • k: ∂(3y² − 3zx)/∂x − ∂(3x² − 3yz)/∂y = −3z + 3z = 0

Answer: div F = 6(x + y + z) and curl F = 0 (F is irrotational, as the gradient of any scalar field).

✓ Answer checked by Claude Opus 5.5 (high effort)

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