Leth∈ ℝ

Theorem. Let𝑈 ⊂ ℝ^𝑛open,𝑓 ∶ 𝑈 → ℝof class𝐶²,a∈ 𝑈. Then

𝑓 (a+h) = 𝑓 (a) +

𝑛

∑𝑖=1

𝜕𝑓

𝜕𝑥_𝑖(a)ℎ_𝑖+1 2

𝑛

∑𝑗=1 𝑛

∑𝑖=1

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a)ℎ_𝑖ℎ_𝑗+ 𝐸(h) wherelim

h→0

𝐸(h)

‖h‖² = 0.

Proof. Leth∈ ℝ^𝑛be of norm small enough to ensure that∀𝑡 ∈ [0, 1],a+ 𝑡h∈ 𝑈. By the previous theorem, there exists𝜃 ∈ (0, 1)such that

𝑓 (a+h) = 𝑓 (a) +

𝑛

∑𝑖=1

𝜕𝑓

𝜕𝑥_𝑖(a)ℎ_𝑖+ 1 2

𝑛

∑𝑗=1 𝑛

∑𝑖=1

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a+ 𝜃h)ℎ_𝑖ℎ_𝑗

Hence

𝐸(h) = 𝑓 (a+h) − 𝑓 (a) −

𝑛

∑𝑖=1

𝜕𝑓

𝜕𝑥_𝑖(a)ℎ_𝑖− 1 2

𝑛

∑𝑗=1 𝑛

∑𝑖=1

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a)ℎ_𝑖ℎ_𝑗

= 1 2

𝑛

∑𝑗=1 𝑛

∑𝑖=1(

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a+ 𝜃h) − 𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a) )ℎ_𝑖ℎ_𝑗

So that

𝐸(h)

‖h‖² = 1 2

𝑛

∑𝑗=1 𝑛

∑𝑖=1(

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a+ 𝜃h) − 𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a) )

ℎ_𝑖ℎ_𝑗

‖h‖² Notice that by continuity of the second order partial derivatives

limh→0

𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a+ 𝜃h) − 𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a) = 𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a) − 𝜕²𝑓

𝜕𝑥_𝑗𝜕𝑥_𝑖(a) = 0 and that ^{|ℎ𝑖ℎ𝑗|}

‖h‖² = ^|ℎ𝑖|

‖h‖

|ℎ_𝑗|

‖h‖ ≤ 1.

Hence lim

h→0

𝐸(h)

‖h‖² = 0. ■