Exercise 2.5.19. Verify that Euler’s formula (nodes – edges + loops = 1) is valid for a square mesh with 10 nodes on each side.

Answer: If the mesh has 10 nodes on each side there are 10 times 10 or 100 nodes in total.

A given side has 9 edges between the 10 nodes on that side. There are 10 times 9 or 90 vertical edges, and 10 times 9 or 90 horizontal edges, for a total of 180 edges.

There are 9 times 9 or 81 independent loops.

We thus have the number of nodes minus the number of edges plus the number of loops equal to 100 – 180 + 81 = 1, in accordance with Euler’s formula.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.