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