Chapter 16.7, Problem 62E

Open and closed boxes Consider the region R bounded by three pairs of parallel planes: ax + by = 0, ax + by = 1, cx + dz = 0, cx + dz = 1, ey + fz = 0, and ey + fz = 1, where a, b, c, d, e, and f are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps.

a.    Find three vectors n₁, n₂. and n₃, each of which is normal to one of the three pairs of planes.

b.    Show that the three normal vectors lie in a plane if their triple scalar product n₁,·(n₂ × n₃) is zero.

c.    Show that the three normal vectors lie in a plane if ade + bcf = 0.

d.    Assuming n₁, n₂, and n₃ lie in a plane P, find a vector N that is normal to P. Explain why a line in the direction of N does not intersect any of the six planes and therefore the six planes do not form a bounded region.

e.    Consider the change of variables u = ax + by, v = cx + dz, w = ey + ft Show that

     $J\left(x,y,z\right)=\frac{\partial \left(u,v,w\right)}{\partial \left(x,y,z\right)}=-ade-bcf$

What is the value of the Jacobian if R is unbounded?