Let S2 = { = (§1. $2. $3) € R³ | || || = 1} be the two-sphere. The stereographic projection o1:S²\{(0,0, –1)} →R?, where R2 is the §1-2 plane or the plane passing through the equator 3 = 0 (not §3 = -1, which is what thetextbook does) is written as1-(§1, §2) with + 3 + = 1.1+ $391($1, §2, §3) =Find an expression for the inverse V1 := 47':R?s2\{(0, 0, 1)} in terms of the coordinatesx = (x1, x2) for the plane R2.Let V := (0, 27) × (0, x) and ý2: VS2 be defined by2(0, 4) = (sin o cos 0, sin o sin 0, cos p).Then 42 := V7:2(V) → V gives another coordinate chart. Find an expression for the change of coordinatesP1 o V2: V → R². [Hint: Using ¢/2 instead of o in the trig functions may simplify the formula.]Let b := 2(0,4) e S² be arbitrary. The two coordinate charts 1 and 2 give rise to two basesa-(b),and(b),(b)} for T,S² . Write the latter in terms of the former.ax1ax2

As per the question we have to find the inverse of the transformation φ1 from R2 to S2 in terms of…

Let S2 = { = (§1. $2. $3) € R³ | || || = 1} be the two-sphere. The stereographic projection o1:S²\{(0,0, –1)} → R², where R2 is the §1-2 plane or the plane passing through the equator 3 = 0 (not §3 = -1, which is what the textbook does) is written as 1 -(§1, §2) with + 3 + = 1. 1+3 Y1($1. $2, §3) = Find an expression for the inverse v1 := 47':R? s2\{(0, 0, 1)} in terms of the coordinates x = (x1, x2) for the plane R2. Let V := (0, 2n) × (0, x) and 2: V S2 be defined by 2(0, 4) = (sin o cos 0, sin o sin 0, cos p). Then 42 := V7: 2(V) → V gives another coordinate chart. Find an expression for the change of coordinates Y1o V2: V → R². [Hint: Using ¢/2 instead of o in the trig functions may simplify the formula.]