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Answer: b
Explanation: Div(P) = Dx(x2yz) + Dy(0) + Dz(xz) = 2xyz + x, which is 2xyz + x. For
different values of x, y, z the divergence of the field varies
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} =