Differential Topology - Computer Graphics and Geometric Modeling - page 508

of a tangent vector as something that specifies a direction in which one can take a
derivative. Here are some properties of these tangent vectors:

(1) Let c U denote the constant map on the neighborhood U with value c. Then

Xc () = 0.

This follows from the identities

() = () = ( ) = ( () + () ) = ()

X c cX 1 cX 1 ◊ 1 c X 1 X 1 2 cX 1 .

U U U U U U U

(2) Let f : U Æ R and V Õ U . Then

( ) =◊ ( ) = () + () ( ) = () Xf V Xf 1 Xf f p X 1 Xf ,

V V

that is, X(f) depends only on the local behavior of f and not on its domain.

(3) Let ( U ,j) be a coordinate neighborhood of p and let j( q ) = (u 1 ( q ),u 2 ( q ),
...,u k ( q )), q ΠU . If

( p , fF Œ

then define

∂ ∂

( ) ()

- 1

() =

f f o jj

(

p .

)

(8.16)

∂ u ∂ x

i i

It is easy to see that ∂/∂u i is a tangent vector at p called the partial derivative with
respect to u i . Furthermore, ∂/∂u 1 , ∂/∂u 2 ,..., ∂/∂u k are a basis for the tangent space
T p ( M k ) (Exercise 8.8.3). (In contrast to other results that hold for C r manifolds also,
the basis property of the ∂/∂u i needs C • , because to show that the ∂/∂u i span, one needs
Lemma 4.6.2. See [BisC64].) It follows that every tangent vector X can be written
uniquely in the form

k

∂

 1

Xa = .

j

∂ u

j

j =

One calls a i the ith component of the tangent vector X.

(4) Let ( V ,y) be another coordinate neighborhood of p and let y ( q ) = (v 1 ( q ),v 2 ( q ),
...,v k ( q )), q Œ V . If we express X with respect to the basis ∂/∂v j , that is, if

k

∂

 1

Xb = ,

j

∂ v

j

j =

then