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To find the value of the divergence of vector field ( mathbf{A} = xy^2 hat{i} + y^3 hat{j} + y^2z hat{k} ) over the cuboid defined by the intervals (0 < x < 1), (0 < y < 1), and (0 < z < 1) using the divergence theorem, we first need to compute the divergence of ( mathbf{A} ).
The divergence of a vector field ( mathbf{A} = Phat{i} + Qhat{j} + Rhat{k} ) is given by:
[ nabla cdot mathbf{A} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]
For ( mathbf{A} = xy^2 hat{i} + y^3 hat{j} + y^2z hat{k} ),
[ P = xy^2, Q = y^3, R = y^2z ]
Computing the partial derivatives,
[ frac{partial P}{partial x} = y^2 ]
[ frac{partial Q}{partial y} = 3y^2 ]
[ frac{partial R}{partial z} = y^2
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