MATH.7:5.3 - Addition survives a coordinate change while length needs a new formula

In X=R^2, let h(x,y)=(x+y,y). Its inverse is r(u,v)=(u-v,v). Transported vector addition coincides with ordinary coordinatewise addition because h is linear.

Now ask for squared Euclidean length, L_X(x,y)=x^2+y^2. The result is a real number whose interpretation stays unchanged, so its output map is the identity. The transported quantity is:

L_Y(u,v)=(u-v)^2+v^2.

The source vector (0,1) has squared length 1 and maps to (1,1). The transported formula still returns 1. The familiar formula u^2+v^2 would return 2.

A length bound L_X(x,y)≤1 consequently becomes (u-v)^2+v^2≤1. It does not become u^2+v^2≤1 through this coordinate map. The example separates an unchanged operation from a quantity whose receiving formula must be constructed. A later physical interpretation of these coordinates uses C.29 for its subject correspondence.