Q. I encountered a problem when I was trying to understand one of the examples in the divergence theorem chapter. In 17.9,the second part of the question of example 2, the example calculates the region E that lies between the closed surfaces S1 and S2. They assigned n = -n1 and n= n2 on S1 and S2 respectively. I was wondering why one of them is negative and the other positive? If using this example as a guide, does it mean that for every region that is bounded inside another region, then the n will always be negative?
A fine question. The student is asking about the text just after Example 2 in Section 17.9.
To use the Divergence Theorem, we have to identify an inside volume and its boundary skin. This inside volume is like the flesh of an avocado, not counting the pit. (Or the white of an egg, not counting the yolk.) The boundary of this volume is not just the outer skin of the avocado, but also the skin of the pit. To orient the surface (which consists of these two pieces) outward, we have to point away from the flesh in both cases (outward means away from the volume). On the skin of the avocado, that’s an outward normal, in the book. On the skin of the pit, that’s a normal pointing away from the flesh — and toward the pit. That’s in the book.