Find the divergence of the field, P = x2yz i + xz k

To find the divergence of the vector field ( mathbf{P} = x^2yz , mathbf{i} + xz , mathbf{k} ), we use the divergence operator, which in three dimensions is given by:

[

nabla cdot mathbf{P} = frac{partial P_x}{partial x} + frac{partial P_y}{partial y} + frac{partial P_z}{partial z}

]

where ( P_x = x^2yz ), ( P_y = 0 ), and ( P_z = xz ).

Now, we compute each term:

1. ( frac{partial P_x}{partial x} = frac{partial}{partial x}(x^2yz) = 2xyz )
2. ( frac{partial P_y}{partial y} = frac{partial}{partial y}(0) = 0 )
3. ( frac{partial P_z}{partial z} = frac{partial}{partial z}(xz) = x )

Now, we add these components together:

[

nabla cdot mathbf{P} = 2xyz + 0 + x = 2xyz + x

]

Thus, the divergence of the field ( mathbf{P} ) is:

[

nabla cdot mathbf{P} =