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By P. Ciarlet

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Extra info for An Intro. to Differential Geometry With Applns to Elasticity

Example text

Since 2m2 is a natural number, it follows that n2 is even using our definition of an even number. This proves our claim. Now that we have proved our statement, we can call it a theorem. There is an obvious companion statement. The square of an odd number is odd. Here is a proof. Let n be an odd number. By definition n=2m+1 for some natural number m. Square both sides of the equation in (2) to get n2=4m2+4m+1. Now rewrite the equation in (3) as n2=2(2m2+2m)+1. Since 2m2+2m is a natural number, it follows that n2 is odd using our definition of an odd number.

This is my chance to thank them for all their kindnesses over the years. The book originated in a course I taught at Heriot-Watt University inherited from my colleagues Richard Szabo and Nick Gilbert. Although the text has been rethought and rewritten, some of the exercises go back to them, and I have had numerous discussions over the years with both of them about what and how we should be teaching. Thanks are particularly due to Lyonell Boulton, Robin Knops and Phil Scott for reading selected chapters, and to Bernard Bainson, John Fountain, Jamie Gabbay, Victoria Gould, Des Johnston and Bruce MacDougall for individual comments.

Result (2) can readily be proved. We use the diagram below. The proof that ॅ=े follows from the simple observation that ॅ+ॆ=ॆ+े. This still leaves (1) and (3). 4. 47 of Euclid. We are given a right-angled triangle. We are required to prove, of course, that a2+b2=c2. Consider the picture below. It has been constructed from four copies of our triangle and two squares of areas a2 and b2, respectively. We claim that this shape is itself actually a square. First, the sides all have the same length a+b.

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