Part 3
Tangent Vectors
Printed version of the lecture Differential Geometry on 7. September 2009
Tommy R. Jensen, Department of Mathematics, KNU
3.1
Overview
Contents
1
Vectors in R3
1
2
Tangent Vectors
2
3
Tangent Space
3
4
Vector Fields
3
5
Conclusion
5
1
3.2
Vectors in R3
Vectors in R3
Example: Force Vectors
3.3
1
Vectors in R3
Example: Velocity Vector
3.4
Vectors in R3
Example: Angular Momentum Vector
3.5
2
Tangent Vectors
Tangent Vectors to R3
Definition 2.1
A tangent vector v p to R3 consists of two points of R3 :
its vector part v,
and its point of application p.
The tangent vector v p is drawn as an arrow from the point p to the point p + v.
If v = (2, 3, 2) and p = (1, 1, 3), then
v p = (2, 3, 2)(1,1,3) starts in (1, 1, 3) and ends in (3, 4, 5),
see Fig. 1.1 in the textbook.
Parallel tangent vectors
Two tangent vectors v p and wq are parallel if and only if v = w.
They are only equal if both v = w and p = q.
3.6
2
3
Tangent Space
Tangent Space
Definition 2.2
Let p be a point of R3 .
Let Tp (R3 ) be the set of all tangent vectors v p with p as their point of application, v ∈ R3 .
Then Tp (R3 ) is called tangent space of R3 at p.
Note: each point of R3 has its own tangent space. They are all different from each other.
To draw a picture of all these different tangent spaces is:
3.7
Tp (R3 ) as a vector space
Addition and scalar multiplication in Tp (R3 )
If v p and w p are two tangent vectors in Tp (R3 ), then
v p + w p = (v + w) p
defines their sum, which also belongs to Tp (R3 ).
If a ∈ R, then
av p = (av) p
defines the scalar multiple of a and v p , this again gives a tangent vector that lies in Tp (R3 ).
Tp (R3 ) is isomorphic to R3
Let p ∈ R3 .
If we define a map from R3 to Tp (R3 ) by
v 7→ v p ,
then this is an isomorphism from
R3
to Tp
(R3 ).
3.8
4
Vector Fields
Vector Fields
Definition 2.3
A vector field on R3 is a function V that assigns to each point p of R3 a tangent vector V (p) to R3 at the
point p.
3.9
3
Force Fields
3.10
Addition and Multiplication with Vector Fields
Defining sum
Let V and W be two vector fields on R3 .
Their sum V +W is a new vector field defined by (again) using the pointwise principle:
(V +W )(p) = V (p) +W (p)
for all
p ∈ R3 .
Defining scalar multiple
Let f : R3 → R be any real-valued function defined on R3 .
The scalar multiple of f and V is the vector field fV such that
( fV )(p) = f (p)V (p)
for all
p ∈ R3 .
3.11
The Natural Frame Field
Definition 2.4
We define three special vector fields on R3 :
U1 (p) = (1, 0, 0) p ,
U2 (p) = (0, 1, 0) p ,
U3 (p) = (0, 0, 1) p ,
for all p in
R3 .(See
Fig. 1.5 of the textbook.)
These three vector fields U1 ,U2 ,U3 together are called the natural frame field on R3 .
3.12
The Euclidean Coordinate Functions of a Vector Field
Lemma 2.5
Let V be any vector field on R3 .
There are three uniquely determined functions
ν1 , ν2 , ν3 : R3 → R,
such that
V = ν1U1 + ν2U2 + ν3U3 .
The functions ν1 , ν2 , ν3 are called the Euclidean coordinate functions of V.
4
3.13
The Euclidean Coordinate Functions of a Vector Field
Proof of Lemma 2.5
Let p be any point in R3 .
Then V (p) is a tangent vector v p at p.
We can write
v = (ν1 (p), ν2 (p), ν3 (p)),
so
V (p)
=
(ν1 (p), ν2 (p), ν3 (p)) p
=
ν1 (p)(1, 0, 0) p + ν2 (p)(0, 1, 0) p + ν3 (p)(0, 0, 1) p
=
ν1 (p)U1 (p) + ν2 (p)U2 (p) + ν3 (p)U3 (p)
=
(ν1U1 )(p) + (ν2U2 )(p) + (ν3U3 )(p)
=
(ν1U1 + ν2U2 + ν3U3 )(p).
So the equality V (p) = (ν1U1 + ν2U2 + ν3U3 )(p) is true for every p in R3 . It follows that V = ν1U1 +
ν2U2 + ν3U3 holds.
3.14
Some Notation
Σ notation
Usually we will write
ΣνiUi = ν1U1 + ν2U2 + ν3U3 .
Sum and scalar product are expressed as
ΣνiUi + ΣwiUi = Σ(νi + wi )Ui
and
f ΣνiUi = Σ( f νi )Ui .
3.15
Differentiable Vector Fields
Definition
A vector field V is differentiable if each of its three Euclidean coordinate functions ν1 , ν2 , ν3 is a differentiable function (smooth, of class C∞ ).
We always assume vector fields are differentiable
From now on, a ‘vector field’ will mean a
‘differentiable vector field’.
3.16
5
Conclusion
And now for something completely different:
END OF THE LECTURE!
Next time:
Directional Derivatives
Operation of a Vector Field
5
3.17
3.18
6