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To determine the divergence of the vector field ( F = 30 i + 2xy j + 5xz^2 k ), we use the formula for divergence in three dimensions:
[
nabla cdot F = frac{partial F_x}{partial x} + frac{partial F_y}{partial y} + frac{partial F_z}{partial z}
]
where ( F_x = 30 ), ( F_y = 2xy ), and ( F_z = 5xz^2 ).
Calculating each term:
1. ( frac{partial F_x}{partial x} = frac{partial (30)}{partial x} = 0 )
2. ( frac{partial F_y}{partial y} = frac{partial (2xy)}{partial y} = 2x )
3. ( frac{partial F_z}{partial z} = frac{partial (5xz^2)}{partial z} = 10xz )
Now summing these results for the divergence:
[
nabla cdot F = 0 + 2x + 10xz
]
Next, we evaluate the divergence at the point ( (1, 1, -0.2) ):
1. Substitute ( x = 1 ), ( y = 1 ), and ( z = -0.2 ) into
Answer: b
Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz2) = 0 + 2x + 10xz = 2x + 10xz
Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.
Alternate/Shortcut: Without calculation, we can easily choose option “0, solenoidal”, as by theory when the divergence is zero, the vector is solenoidal. “0, solenoidal” is the only one which is satisfying this condition.