## Linear Algebra and Its Applications, Exercise 2.5.19

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 .

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