042-Fundamentals-of-Physics-II-Electromagnetism-Optics-and-Quantum-Mechanics-R.-Shankar-Edisi-1-2016 · Problem 15.10.1

Electromagnetism and Relativity

Physics — Shankar · tensor-invariant-verification

Problem

Verify from Eqns. 15.158 to 15.164 that the trace

S = T_{xx} + T_{yy} + T_{zz}

is invariant under rotations around the

z

-axis, i.e., that

S = T_{xx} + T_{yy} + T_{zz} = T'_{xx} + T'_{yy} + T'_{zz} = S'

Intuitive explanation

The trace of a tensor captures a fundamental geometric property—the sum of its diagonal elements—that shouldn't depend on which way you're looking at it. When you rotate your coordinate system around the $z$ -axis, the tensor components change according to a rotation matrix, but the trace must remain the same because it's measuring something intrinsic to the physical quantity itself. The key is to substitute the rotation transformation rules into the new diagonal components and show that the cross terms cancel out, leaving you with the original trace.

Intuitive explanation

Browse the problem library