By Frederick S. Woods

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**Example text**

Then x 2 −π 2 = 2δπ +δ2 , and an easy calculation shows that this has modulus less than 10−100 if δ has modulus less than 10−101 . So we will be all right if we take the ﬁrst 101 digits of π after the decimal point. More generally, however accurate we wish our estimate of π 2 to be, we can achieve this accuracy if we are prepared to make x a suﬃciently good approximation to π . In mathematical parlance, the function f (x) = x 2 is continuous at π . Let us try to say this more symbolically. The statement “x 2 = π 2 to within an accuracy of ” means that |x 2 −π 2 | < .

The three most commonly used models of hyperbolic geometry are called the half-plane model, the disk model, and the hyperboloid model. The half-plane model is the one most directly associated with the group PSL(2, R). The set in question is the upper half-plane of the complex numbers C, that is, the set of all complex numbers z = x + yi such that b y > 0. Given a matrix ( ac d ), the corresponding transformation is the one that takes the point z to the point (az +b)/(cz +d). ) The condition ad − bc = 1 can be used to show that the transformed point will still lie in the upper half-plane, and also that the transformation can be inverted.

The following notation is common: if S is the set and f is the function, then the total amount of f in S, known as the integral, is written S f (x) dx. Here, x denotes a typical element of S. If, as in the density example, the elements of S are points (x, y, z), then vector notation such as S f (x) dx can be used, though often it is not and the reader is left to deduce from the context that an ordinary “x” denotes a vector rather than a real number. We have been at pains to distinguish integration from antidiﬀerentiation, but a famous theorem, known as the fundamental theorem of calculus, asserts that the two procedures do, in fact, give the same answer, at least when the function in question has certain continuity properties that all “sensible” functions have.

### A course in mathematics,: For students of engineering and applied science, by Frederick S. Woods

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