Question

I need Help with this physics question please !

Answer 1

I'll denote vectors in boldface. So the given vector field is

B = x² x - y² y + 3z z

Compute the divergence:

∇ • B = (∂/∂x x + ∂/∂y y + ∂/∂z z) • (x² x - y² y + 3z z)

∇ • B = ∂/∂x [x²] + ∂/∂y [-y²] + ∂/∂z [3z]

∇ • B = 2x - 2y + 3

Compute the curl:

∇ × B = (∂/∂x x + ∂/∂y y + ∂/∂z z) × (x² x - y² y + 3z z)

∇ × B = ∂/∂x [x²] (x × x) + ∂/∂x [-y²] (x × y) + ∂/∂x [3z] (x × z)

… … … + ∂/∂y [x²] (y × x) + ∂/∂y [-y²] (y × y) + ∂/∂y [3z] (y × z)

… … … + ∂/∂z [x²] (z × x) + ∂/∂z [-y²] (z × y) + ∂/∂z [3z] (z × z)

∇ × B = 0

(since each partial derivative not along the main diagonal vanishes, and for any vector a we have a × a = 0)