Covariant basis

Description

I consider

∂→Ei∂Xj.

By using the relation

→Ei=∑j∂xj∂Xi→ej,

this is equal to

∂∂Xj(∑l∂xl∂Xi→el)=∑l∂∂Xj(∂xl∂Xi)→el+∑l∂xl∂Xi∂→el∂Xj.

Since the reference Cartesian basis vectors are constant, the second term disappears and I am left with

∑l∂∂Xj(∂xl∂Xi)→el=∑l∂∂Xj(∂xl∂Xi)(∑k∂Xk∂xl→Ek)=∑kl∂∂Xj(∂xl∂Xi)∂Xk∂xl→Ek=∑k{∑l∂∂Xj(∂xl∂Xi)∂Xk∂xl}→Ek=∑kΓkij→Ek,

where Γkij is the Christoffel symbol of the second kind:

Γkij≡∑l∂∂Xi(∂xl∂Xj)∂Xk∂xl,

and by using the relation between the symbol and the scale factors:

2HkHkΓkij=−∂∂Xk(δijHiHj)+∂∂Xi(δjkHjHk)+∂∂Xj(δkiHkHi),

the change of the basis vector is given by

∑k12HkHk{−∂∂Xk(δijHiHj)+∂∂Xj(δkiHkHi)+∂∂Xi(δjkHjHk)}→Ek=−∑kδijHj2HkHk∂Hi∂Xk→Ek−∑kδijHi2HkHk∂Hj∂Xk→Ek+1Hi∂Hi∂Xj→Ei+1Hj∂Hj∂Xi→Ej,

and in summary:

∂→Ei∂Xj=−∑kδijHj2HkHk∂Hi∂Xk→Ek−∑kδijHi2HkHk∂Hj∂Xk→Ek+1Hi∂Hi∂Xj→Ei+1Hj∂Hj∂Xi→Ej.

This relation is written down explicitly below for an intuitive understanding.

(∂→E1∂X1∂→E2∂X1∂→E3∂X1∂→E1∂X2∂→E2∂X2∂→E3∂X2∂→E1∂X3∂→E2∂X3∂→E3∂X3)=−(∑kH1HkHk∂H1∂Xk→Ek000∑kH2HkHk∂H2∂Xk→Ek000∑kH3HkHk∂H3∂Xk→Ek)+(01H1∂H1∂X2→E1+1H2∂H2∂X1→E21H1∂H1∂X3→E1+1H3∂H3∂X1→E3sym.01H2∂H2∂X3→E2+1H3∂H3∂X2→E3sym.sym.0).

Example

The changes of the basis vectors

(∂→E1∂X1∂→E2∂X1∂→E3∂X1∂→E1∂X2∂→E2∂X2∂→E3∂X2∂→E1∂X3∂→E2∂X3∂→E3∂X3)

read as follows.

Cylindrical coordinates

(01X1→E201X1→E2−X1→E10000).

Rectilinear coordinates

(−1H1∂H1∂X1→E1000−1H2∂H2∂X2→E2000−1H3∂H3∂X3→E3).

Application

(−1H1∂H1∂X1→E11H2∂H2∂X1→E201H2∂H2∂X1→E2−H2H1H1∂H2∂X1→E10000).