### D.3 Coordinate transformation derivation

This note derives the coordinate transformation formulae of chapter 1.4.2.

According to the total differential formula from calculus:

This formula is used to transform the first order derivatives of .

Differentiating once more, using the product rule of differentiation and again the total differential formulae for the first factor of the product:

If you plug the formula for the second order derivatives above into the left hand side of the original partial differential equation and rearrange, you get the transformed equation.

The coefficients of the transformed matrix and the transformed right hand side can be read off from the above expression.