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

Consider the top face (ABFE) for which the normal is $\hat k$so that the surface integral is $\int F_z dydx$. On this face $z=1$and $F_z=y$. The contribution to the surface integral from this face is

\begin{displaymath}\int_0^1 dx\int_0^1 ydy = \frac{1}{2}\end{displaymath}

For the bottom face (DOGC) the normal is along $-\hat k$and $z=0$. This gives $F_z=0$so that the integral vanishes.
For the face EFGO the normal is along $-\hat\imath$so that the surface integral is $-\int F_xdydz$. On this face $x=0$giving $F_x=0$. The surface integral is zero. For the front face ABCD, the normal is along $\hat\imath$and on this face $x=1$giving $F_x= 4z$. The surface integral is

\begin{displaymath}\int_0^1 dy\int_0^1 4zdz = 2\end{displaymath}

Adding the six contributions above, the surface integral is $3/2$consistent with the divergence theorem.

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