### D.4 2D coordinate transformation derivation

This note gives a more detailed description where the expressions (1.12) for the coefficients , , and comes from.

In terms of the notations for the general case, you have

and

The introduction noted that the new coefficients can be found from

As an example, let’s find the value for . In terms of the notations in the general case, is the 1,1 element of matrix : . To get , put in

That turns and into , or for short:

If we write out the four terms of the double sum explicitly, that becomes:

Now note that by definition , , , , and , and you get the expression for claimed:

The expressions for , and may be verified similarly.