Differential-Geometry of Manifolds Functions from Rn-to-Rm, guaranteed and verified 100% Pass

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Functions from ℝ𝑛 to ℝ𝑚

ℝ𝑛 = {(𝑥1 , … , 𝑥𝑛 )| 𝑥𝑖 ∈ ℝ, 𝑖 = 1, … , 𝑛}

ℝ𝑛 is a vector space with standard basis {𝑒⃗1 , … , 𝑒⃗𝑛 } where
𝑒⃗𝑖 = < 0, 0, 0, 1, 0, … , 0 > (1 in the 𝑖𝑡ℎ place). The standard norm on ℝ𝑛 is
given by:

‖𝑥⃗ ‖ = √𝑥12 + ⋯ + 𝑥𝑛2 , where 𝑥⃗ = < 𝑥1 , … , 𝑥𝑛 >.



We can define a distance on ℝ𝑛 by:

𝑑 (𝑥⃗, 𝑦⃗) = ‖𝑥⃗ − 𝑦⃗‖ = √(𝑥1 − 𝑦1 )2 + ⋯ + (𝑥𝑛 − 𝑦𝑛 )2 .



Def. If 𝑓: 𝐴 ⊆ ℝ𝑛 → ℝ𝑚 , we say that 𝑓 is continuous at 𝒂⃗⃗ ∈ 𝑨 if for all
𝜖 > 0 there exists a 𝛿 > 0 such that if ‖𝑥⃗ − 𝑎⃗‖ < 𝛿, then
‖𝑓 (𝑥⃗ ) − 𝑓(𝑎⃗)‖ < 𝜖.



Def. If 𝑓: 𝐴 ⊆ ℝ𝑛 → ℝ and 𝑎⃗ ∈ 𝐴, then we define the 𝒊𝒕𝒉 partial derivative of
𝑓 at 𝑎⃗ as:
𝜕𝑓 𝑓(𝑎1 , … , 𝑎𝑖 + ℎ, … , 𝑎𝑛 ) − 𝑓(𝑎1 , … , 𝑎𝑛 )
(𝑎⃗) = lim
𝜕𝑥𝑖 ℎ→0 ℎ
as long as the limit exists.

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Def. If 𝑓: 𝐴 ⊆ ℝ𝑛 → ℝ𝑚 , we say that f is differentiable at 𝒂
⃗⃗ ∈ 𝑨 if there
exists a linear transformation 𝜆: ℝ𝑛 → ℝ𝑚 such that:

‖𝑓(𝑎⃗ + ℎ⃗⃗) − 𝑓(𝑎⃗) − 𝜆(ℎ
⃗⃗)‖
lim = 0.
⃗ℎ⃗→0 ⃗⃗
‖ℎ‖
In this case, we say 𝐷𝑓 (𝑎⃗) = 𝜆.

Let 𝑓: 𝐴 ⊆ ℝ𝑛 → ℝ𝑚 , then we can write:

𝑓(𝑥1 , … , 𝑥𝑛 ) = (𝑓1 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ), 𝑓2 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ), … , 𝑓𝑚 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ))
where 𝑓𝑖 : ℝ𝑛 → ℝ.

𝜕𝑓𝑖
Theorem: If 𝑓: ℝ𝑛 → ℝ𝑚 is differentiable at 𝑎⃗ ∈ ℝ𝑛 , then (𝑎⃗⃗) exists
𝜕𝑥𝑗
for 1 ≤ 𝑖 ≤ 𝑚 , 1 ≤ 𝑗 ≤ 𝑛 , and
𝜕𝑓1 𝜕𝑓1
⋯
𝜕𝑥1 𝜕𝑥𝑛
𝐷𝑓 (𝑎⃗) = ⋮ ⋮
𝜕𝑓𝑚 𝜕𝑓𝑚
⋯
( 𝜕𝑥1 𝜕𝑥𝑛 )
𝜕𝑓𝑖
where is evaluated at 𝑎⃗.
𝜕𝑥𝑗


Def. 𝐷𝑓 (𝑎⃗ ) is called the Jacobian matrix of 𝑓 at 𝑎⃗. So if 𝐷𝑓 (𝑎⃗) exists, then all
𝜕𝑓𝑖 𝜕𝑓𝑖
of the partial derivatives, , exist at 𝑎⃗. The converse is not true: all of
𝜕𝑥𝑗 𝜕𝑥𝑗
existing at 𝑎⃗ does not imply 𝐷𝑓 (𝑎⃗ ) exists.


Theorem (Chain Rule): If 𝑓: ℝ𝑛 → ℝ𝑚 is differentiable at 𝑎⃗ ∈ ℝ𝑛 , and
𝑔: ℝ𝑚 → ℝ𝑝 is differentiable at 𝑓(𝑎⃗), then 𝑔 ∘ 𝑓: ℝ𝑛 → ℝ𝑝
is differentiable at 𝑎⃗ and 𝐷 (𝑔 ∘ 𝑓 )(𝑎⃗ ) = 𝐷𝑔(𝑓 (𝑎⃗ )) ∘ 𝐷𝑓 (𝑎⃗ ).