Let M be the set of points in the space where the second and third components of the S² (unit sphere) manifold are positive. (M is a 2-dimensional manifold)

, shaped coordinate system is given for manifold M. (So is a map.)

For function defined as ,

( Here, the usual manifold structure of ² will be discussed. 's are taken as coordinate functions of and 's as canonical coordinate functions of ² )

a) Find the matrix.

b) Specify the matrix by taking .

c) Since and , find the vector as the linear combination of base

R : M&M $(P1, P2, P3) = (P3-P1) (M.) F F: MR? F(P1:P2.P3) = (P + P2. 2p1 + 3p3) 12 JF (JF) 1 V3 p= (0, 2 2 = 3 Ip +2 I- 2 F.pl Up [F(): j= 1, 2