D.4 Coordinate transformation derivation

This note derives the coordinate transformation formulae of chapter 18.7.2.

According to the total differential formula from calculus:

$$\frac{\partial u}{\partial x_i} = \sum_{k=1}^n \frac{\partial u}{\partial \xi_k} \frac{\partial \xi_k}{\partial x_i}$$

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

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

$$\frac{\partial^2 u}{\partial x_i\partial x_j} = \sum_{k=... ... \frac{\partial^2 \xi_k}{\partial x_i\partial x_j} \right]$$

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.

$$\sum_{k=1}^n \sum_{l=1}^n \left( \sum_{i=1}^n \sum_{j=... ..._i\partial x_j} \right) \frac{\partial u}{\partial \xi_k}$$

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