Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function

Consider the surface integral from the six faces individually. For the face AEOD, the normal is along $-\hat\jmath$. On this face $y=0$so that $\vec F = 4xz\hat\imath$. Since $\hat\imath\cdot\hat\jmath =0$, the integrand is zero. For the surface BFGC, the normal is along $\hat\jmath$and on this face $y=1$. On this face the vector field is $\vec F = 4xz\hat\imath -\hat\jmath + z\hat k$. The surface integral is

\begin{eqnarray*} \int \vec F\cdot\vec{dS} &=& \int \vec F\cdot\hat\jmath dxdz\\ &=& -\int_0^1dx\int_0^1 dz= -1 \end{eqnarray*}

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