Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 16 / b / i / Solution

The PDF diagram specifies the prism asIts triangular horizontal cross-section has area four. Calculate the outward flux on each face using its outward normal vector.

On $x=0$, the normal is $-{\mathbf{e}}_x$ and $F_x=0$, so the flux is zero. On $y=0$, the normal is $-{\mathbf{e}}_y$, givingOn the sloping vertical face $y+2x=4$, the outward normal is proportional to $(2,1,0)$. Both $F_x$ and $F_y$ vanish there, while $F_z$ is tangent to the face, so the flux is zero.

On the bottom $z=0$, the normal is $-{\mathbf{e}}_z$ and $F_z=1$, so the flux is minus the base area, namely $-4$. On the top $z=1$, $F_z=0$, so the flux is zero. Thus all five face values and their sum are