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To apply the divergence theorem, we first need to compute the divergence of the vector field ( mathbf{A} = xy^2 mathbf{i} + y^3 mathbf{j} + y^2z mathbf{k} ).
The divergence ( nabla cdot mathbf{A} ) is given by:
[
nabla cdot mathbf{A} = frac{partial (xy^2)}{partial x} + frac{partial (y^3)}{partial y} + frac{partial (y^2z)}{partial z}
]
Calculating each term individually:
1. ( frac{partial (xy^2)}{partial x} = y^2 )
2. ( frac{partial (y^3)}{partial y} = 3y^2 )
3. ( frac{partial (y^2z)}{partial z} = y^2 )
Adding these together, we get:
[
nabla cdot mathbf{A} = y^2 + 3y^2 + y^2 = 5y^2
]
Next, to apply the divergence theorem, we need to integrate the divergence over the volume of the cuboid defined by ( 0 < x < 1 ), ( 0 < y < 1 ), and ( 0 < z < 1 ):
Answer: c
Explanation: A cuboid has six faces. ∫∫A.ds = ∫∫Ax=0 dy dz + ∫∫Ax=1 dy dz + ∫∫Ay=0 dx dz+ ∫∫Ay=1 dx dz + ∫∫Az=0 dy dx + ∫∫Az=1 dy dx. Substituting A and integrating we get (1/3)+ 1 + (1/3) = 5/3.