Vector Calculus | Part IA, 2006

Consider the vector field F(x)=((3x3x2)y,(y32y2+y)x,z21)\mathbf{F}(\mathbf{x})=\left(\left(3 x^{3}-x^{2}\right) y,\left(y^{3}-2 y^{2}+y\right) x, z^{2}-1\right) and let SS be the surface of a unit cube with one corner at (0,0,0)(0,0,0), another corner at (1,1,1)(1,1,1) and aligned with edges along the xx-, yy - and zz-axes. Use the divergence theorem to evaluate

I=SFdSI=\int_{S} \mathbf{F} \cdot d \mathbf{S}

Verify your result by calculating the integral directly.

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