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.]
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.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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 the
textbook does) is written as
1
-(§1, §2) with + 3 + = 1.
1+ $3
91($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, 27) × (0, x) and ý2: 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
P1 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 bases
a
-(b),
and
(b),
(b)} for T,S² . Write the latter in terms of the former.
ax1
ax2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc8424dd-54af-4eb1-809a-68981165dbdf%2Ffed1248d-0653-458a-b4ce-af2b928533e2%2Fpnexmu_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 the
textbook does) is written as
1
-(§1, §2) with + 3 + = 1.
1+ $3
91($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, 27) × (0, x) and ý2: 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
P1 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 bases
a
-(b),
and
(b),
(b)} for T,S² . Write the latter in terms of the former.
ax1
ax2
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