# Applied Mathematics: v. 1 by L. Bostock, F.S. Chandler

By L. Bostock, F.S. Chandler

Best mathematics books

The Electric Motor and its Applications

In contrast to another reproductions of vintage texts (1) we haven't used OCR(Optical personality Recognition), as this results in undesirable caliber books with brought typos. (2) In books the place there are photographs equivalent to snap shots, maps, sketches and so forth we now have endeavoured to maintain the standard of those photos, so that they characterize effectively the unique artefact.

Extra info for Applied Mathematics: v. 1

Example text

Our proof is based on a process of removing an edge or an edge and a vertex from a connected graph so that the resulting graph is still connected. We will then show that V - E + F does not change during this process, and finally we show that through repeated use of this process we arrive at a graph for which V - E + F == 2. This will clearly show that Euler's equation holds for the original graph. The process consists of doing one of two things: either removing an edge, but leaving both of its endpoints, or removing an edge and one of its endpoints.

When we remove a I-valent vertex and its edge, we decrease V and E 28 MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM by one while leaving F unchanged, thus V - E + F is unchanged. When we remove an edge and leave its two endpoints, we decrease E by one. Since the removed edge is on two different faces (one inside the circuit and one outside the circuit), and since these two faces become one face after the removal, F is decreased by one. Again, V - E + F is left unchanged. Now observe what happens when we do this process repeatedly to a given planar graph.

Your results should correspond to the last four lines of Table 1. v E F TETRAHEDRON 4 6 4 CUBE 8 12 6 OCTAHEDRON 6 12 8 DODECAHEDRON 20 30 12 ICOSAH E DRO N 12 30 20 TABLE 1 You should not get the impression that we have proved the existence of any regular polyhedra. What we have done is proved that if a regular polyhedron of a given type exists, then it must have a certain number of vertices, edges and faces. We have alnzost given a proof that there can be at most five regular polyhedra. Many authors present an argument like ours as if it were a proof, but it is not.