Lecture 2 13 Figure 1.10. The sphere as a union of graphs. What good is all this? What benefit do we gain from representing the torus, or any other surface, by an equation? Of course, it allows us to plug the equation into a computer and look at pretty pictures of our surface, but what we are really after is coordinates on our surface. After all, the surface is a two-dimensional affair, and so we should be able to describe its points using just two variables, but the equations we obtain are written in three variables. To address this, we first backtrack a bit and discuss graphs of functions. Recall that given a function f : R2 → R, the graph of f is graph f = { (x, y, z) ∈ R3 | z = f(x, y) }. If f is ‘nice’, its graph is a ‘nice’ surface sitting in R3. Of course, most surfaces cannot be represented globally as the graph of such a function the sphere, for instance, has two points on the z-axis, and hence we require at least two functions to describe it in this manner. In fact, more than two functions are required if we adopt this approach. The unit sphere is given as the solution set of x2+y2 +z2 = 1, so we can write it as the union of the graphs of f1 and f2, where f1(x, y) = 1 − x2 − y2, f2(x, y) = − 1 − x2 − y2.

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