... in (2. 31):d 2 xµdµ 2 + µνσdxνdµdxσdµ=dµdλ 2 f (λ)dµdλ−d 2 µdλ 2 dxµdµ. (2. 32) This has the same form as (2. 31) but involves a different function of µ on the right-hand ... =QPgµνx(λ)dxµdλdxνdλ1 /2 dλ. (2. 38)In the space of three-dimensional Euclidean geometry, the squared element ofdistance expressed in Cartesian coordinates is ds 2 = (dx1) 2 + (dx 2 ) 2 + (dx3) 2 , ... manifold. 22 Geometry Figure 2. 8. Same as figure 2. 7, but using different coordinates.The following examples illustrate, in terms of two-dimensional manifolds,some of the important ideas. Figure 2. 7(a)...