1
Tangent Spaces
Let 𝑀 ⊆ ℝ𝑛 be a 𝑘-dimensional manifold and ⃗Φ
⃗⃗ a parameterization where
⃗Φ
⃗⃗ : 𝑈 ⊆ ℝ𝑘 → 𝑀 ⊆ ℝ𝑛 and ⃗Φ ⃗⃗ (𝑎) = 𝑥 ∈ 𝑀, then:
𝐷Φ⃗⃗⃗ (𝑎): ℝ𝑘𝑎 → ℝ𝑛𝑥 .
𝑘
⃗⃗⃗ (𝑎)(ℝ𝑎 ) = 𝑇𝑥 (𝑀) the tangent space of 𝑀 at 𝑥.
Def. We call 𝐷Φ
⃗⃗⃗ .
Note: this definition does not depend on the parameterization Φ
Def. A tangent vector to a manifold, 𝑀, at a point 𝑝 ∈ 𝑀, is the tangent vector
at 𝑝 of a curve in 𝑀 passing through 𝑝.
, 2
Ex. Find a description of the tangent plane to the torus in ℝ4 given by:
⃗Φ
⃗⃗ (𝑢, 𝑣) = (cos 𝑢 , sin 𝑢 , cos 𝑣 , sin 𝑣 )
𝜋 𝜋
at the point where: (𝑢, 𝑣 ) = ( , ).
6 4
− sin 𝑢 0
⃗⃗⃗ (𝑢, 𝑣) = ( cos 𝑢
𝐷Φ
0
)
0 − sin 𝑣
0 cos 𝑣
1
− 0
2
√3
0
⃗⃗⃗ (𝜋 , 𝜋) =
𝐷Φ
2
.
6 4 √2
0 −
2
√2
( 0 2 )
The tangent space is spanned by the image of < 1,0 > and
⃗⃗⃗ (𝜋 , 𝜋).
< 0,1 > under 𝐷Φ
6 4
1
− 0 1
2
√3
−
2
0
⃗⃗⃗ (𝜋 , 𝜋)) (1) =
(𝐷Φ
2
(
1
)=
√3
6 4 0 0 −
√2 0 2
2 0
√2 ( 0 )
( 0 2 )
Tangent Spaces
Let 𝑀 ⊆ ℝ𝑛 be a 𝑘-dimensional manifold and ⃗Φ
⃗⃗ a parameterization where
⃗Φ
⃗⃗ : 𝑈 ⊆ ℝ𝑘 → 𝑀 ⊆ ℝ𝑛 and ⃗Φ ⃗⃗ (𝑎) = 𝑥 ∈ 𝑀, then:
𝐷Φ⃗⃗⃗ (𝑎): ℝ𝑘𝑎 → ℝ𝑛𝑥 .
𝑘
⃗⃗⃗ (𝑎)(ℝ𝑎 ) = 𝑇𝑥 (𝑀) the tangent space of 𝑀 at 𝑥.
Def. We call 𝐷Φ
⃗⃗⃗ .
Note: this definition does not depend on the parameterization Φ
Def. A tangent vector to a manifold, 𝑀, at a point 𝑝 ∈ 𝑀, is the tangent vector
at 𝑝 of a curve in 𝑀 passing through 𝑝.
, 2
Ex. Find a description of the tangent plane to the torus in ℝ4 given by:
⃗Φ
⃗⃗ (𝑢, 𝑣) = (cos 𝑢 , sin 𝑢 , cos 𝑣 , sin 𝑣 )
𝜋 𝜋
at the point where: (𝑢, 𝑣 ) = ( , ).
6 4
− sin 𝑢 0
⃗⃗⃗ (𝑢, 𝑣) = ( cos 𝑢
𝐷Φ
0
)
0 − sin 𝑣
0 cos 𝑣
1
− 0
2
√3
0
⃗⃗⃗ (𝜋 , 𝜋) =
𝐷Φ
2
.
6 4 √2
0 −
2
√2
( 0 2 )
The tangent space is spanned by the image of < 1,0 > and
⃗⃗⃗ (𝜋 , 𝜋).
< 0,1 > under 𝐷Φ
6 4
1
− 0 1
2
√3
−
2
0
⃗⃗⃗ (𝜋 , 𝜋)) (1) =
(𝐷Φ
2
(
1
)=
√3
6 4 0 0 −
√2 0 2
2 0
√2 ( 0 )
( 0 2 )
