Answers to quiz 7

1. The gradients for these surfaces are n1 = (yz,xz,xy), n2 = (1,1,1), and n3 = (2x,2y,−4z). Since n1·n3 = 0 for any (x,y,z), surfaces 1 and 3 (and only they) are mutually perpendicular at each point.
2. One of the volumes is a pyramid with corners at (−1,1,1), (1,−1,1), (1,1,−1), (1,1,1). Its volume is a3/6 (a=2 is the edge size), and the cube volume is a3, so the other part has a volume of 5a3/6, and the ratio of the two volumes is 5:1

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