Differential Geometry of Manifolds Differentiable Maps Between Manifolds, guaranteed and verified 100% Pass

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Differentiable Maps Between Manifolds

Def. Let 𝑀 and 𝑁 be differentiable manifolds. A continuous function,
𝑓: 𝑀 → 𝑁, is said to be differentiable if for any coordinate chart
𝑦: 𝑉 → ℝ𝑚 on 𝑁 and any chart 𝑥: 𝑈 → ℝ𝑛 on 𝑀, the map:
𝑦 ∘ 𝑓 ∘ 𝑥 −1 : 𝑥(𝑈 ∩ 𝑓 −1 (𝑉)) ⊆ ℝ𝑛 → 𝑦(𝑉) ⊆ ℝ𝑚 is differentiable.




𝑈 𝑓 −1 (𝑉) ∩ 𝑈 𝑉
𝑓
𝑀 𝑁




𝑥
𝑦




𝑥(𝑈) 𝑦(𝑉)
𝑦 ∘ 𝑓 ∘ 𝑥 −1




𝑥(𝑓 −1 (𝑉) ∩ 𝑈)

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In practice how do we differentiate a function with respect to different
coordinate charts?

Ex. Consider the unit sphere with the atlas defined by stereographic
projections: 𝐴 = {(𝑆 2 − (0, 0, 1), 𝜋𝑁 ), (𝑆 2 − (0, 0, −1), 𝜋𝑆 } given
𝑥 𝑦
by: (𝑢, 𝑣) = 𝜋𝑁 (𝑥, 𝑦, 𝑧) = ( ,
1−𝑧 1−𝑧
)
𝑥 −𝑦
(𝑢̅, 𝑣̅ ) = 𝜋𝑆 (𝑥, 𝑦, 𝑧) = (1+𝑧 , 1+𝑧 )
Take the function 𝑓: 𝑆 2 → ℝ by 𝑓 (𝑥, 𝑦, 𝑧) = 𝑧.

𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓
a. Find , , ̅ , 𝜕𝑣̅
𝜕𝑢 𝜕𝑣 𝜕𝑢
𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝜕𝑓
b. Find formulas that relate to and , and to and .
𝜕𝑢 ̅
𝜕𝑢 𝜕𝑣̅ 𝜕𝑣 ̅
𝜕𝑢 𝜕𝑣̅
c. Consider the point on the sphere in Cartesian coordinates given by
1 √2 1
(2 , , ). Find the coordinates in (𝑢, 𝑣) and (𝑢̅, 𝑣̅ ).
2 2
1 √2 1
d. Show that the relationship in part b works for the point ( , , ).
2 2 2


𝑥 𝑦 𝑥 𝑦
a. Since 𝑢 = and 𝑣 = from (𝑢, 𝑣 ) = 𝜋𝑁 (𝑥, 𝑦, 𝑧) = ( , )
1−𝑧 1−𝑧 1−𝑧 1−𝑧

2𝑢 2𝑣 𝑢2 +𝑣 2 −1
𝜋𝑁−1 (𝑢, 𝑣) =( , , )
𝑢2 +𝑣 2 +1 𝑢2 +𝑣 2 +1 𝑢2 +𝑣 2 +1
(from an earlier homework assignment)

2𝑢
That is 𝑥=
𝑢2 +𝑣 2 +1
2𝑣
𝑦=
𝑢2 +𝑣 2 +1

𝑢2 +𝑣 2 −1
𝑧= .
𝑢2 +𝑣 2 +1