Mathematical Tools for Physics - Department of Physics - University

12—Tensors 324

using thexˆ,yˆ,zˆbasis.

Ans:∂Tij/∂xjin coordinate basis.

12.23 ComputedivT in cylindrical coordinates using both the coordinate basis and the usual unit

vector(r,ˆφ,ˆzˆ)basis. This is where you start to see why the coordinate basis has advantages.

12.24 Show thatgijgj=δi.

12.25 If you know what it means for vectors at two different points to be parallel, give a definition for
what it means for two tensors to be parallel.

12.26 Fill in the missing steps in the derivation following Eq. (12.24), and show that the alternating
tensor is unique up to a factor.

12.27 The divergence of a vector field computed in a coordinate basis isdivF~ =∂Fi/∂xi. The

divergence computed in standard cylindrical coordinates is Eq. (9.15). Show that these results agree.

12.28 From Eq. (12.31) verify the stated properties of theω= 0solution.