By Luther Pfahler Eisenhart

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**Extra resources for An introduction to differential geometry with use of the tensor calculus**

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6, the function ¢( a, b, /) is an increasing function of /, for fixed a, b. If two triangles had the same values of a, b, c, but different values of /, then ¢( a, b, /1) would be equal to ¢(a, b, /2), for /1 =f. /2, and that would contradict the increasing nature of the function. We conclude that the triangles have the same value of / and 0 hence that they are congruent by the SAS criterion. We sometimes write the inverse of the equation c = ¢( a, b, /) as / = 'IjJ(a, b, c), so that 'IjJ(a, b, c) is the angle opposite to the side c, and 'IjJ(a, b, c) = 'IjJ(b, a, c).

For example: (3) A nonempty set that is bounded below has a greatest lower bound. (4) Every bounded monotonic sequence has a limit. (5) Every continuous function defined on a bounded closed interval attains a maximum (hence also a minimum). (6) Every continuous function has the intermediate-value property. Exercises. In the following, IF is always a complete (hence Archimedean) ordered field, as defined above. 1. Prove that if a is any positive element of IF, there are rational numbers x such that 0 < x < a.

Proof: Let the vertices be A, B, G, D, with right angles at A, B, G, as in Fig. 8a. We must prove that the angle at D is ~ 90 0 and also that IABI ~ IDC! and IBC! ~ IADI. Draw the diagonal BD. The sum of the six angles of the two resulting triangles is equal to the sum of the four angles of the quadrilateral. 12 applied to those triangles, we see that the sum of the four angles of the quadrilateral is ~ 360 0 , hence the angle at D is ~ 90 0 • To prove the second part, let E be the midpoint of BG and construct the perpendicular to BG at E, and let F be the intersection of that perpendicular with AD, as in Fig.