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Unformatted text preview: 2.1. COVECTORS AND RIEMANNIAN METRIC 27 1. 2.1.2
• Diﬀerential of a Function
Deﬁnition 2.1.3 Let M be a manifold and p ∈ M be a point in M. The
∗
space T p M dual to the tangent space T p M at p is called the cotangent
space.
Deﬁnition 2.1.4 Let M be a manifold and f : M → R be a real valued
smooth function on M. Let p ∈ M be a point in M. The diﬀerential
∗
d f ∈ T p M of f at p is the linear functional • d f : Tp → R
deﬁned by
d f (v) = v p ( f ) . • In local coordinates x j the diﬀerential is deﬁned by
n d f (v) = v j ( x)
j=1 In particular, ∂f
∂x j ∂
∂f
= j.
∂x j
∂x df
Thus
dxi ∂
= δij .
j
∂x and
dxi (v) = vi .
∗
• The diﬀerentials {dxi } form a basis for the cotangent space T p M called the
coordinate basis. • Therefore, every linear functional has the form
n α= a j dx j .
j =1 diﬀgeom.tex; April 12, 2006; 17:59; p. 30 28 CHAPTER 2. TENSORS
• That is why, the linear functionals are also called diﬀerential forms, or
1forms, or covectors, or covariant vectors. • Deﬁnition 2.1.5 A covector ﬁeld α is a diﬀerentiable assignment of
∗
a covector α p ∈ T p M to each point p of a manifold. This means that
the components of the covector ﬁeld are diﬀerentiable functions of local
coordinates. • Therefore, a covector ﬁeld has the form
n α= a j ( x)dx j
j =1 j
j
• Under a change of local coordinates xα = xα ( xβ ) the diﬀerentials transform
according to
n
j
∂ xα i
j
dxα =
dx .
i
∂ xβ β
i=1 • Therefore, the components of a covector transform as
n aα =
i
j =1 2.1.3 j
∂ xα β
a.
i
∂ xβ j Inner Product • Let E be a ndimensional real vector space.
• The inner product (or scalar product) on E is a symmetric bilinear nondegenerate functional on E × E , that is, it is a map , : E × E → R such
that
1. ∀v, w ∈ E ,
v, w = w, v
2. ∀v, w, u ∈ E , ∀a, b ∈ R
av + bu, w = a v, w + b u, w
3. ∀v ∈ E ,
if v, w = 0, ∀w ∈ E , then v = 0 .
diﬀgeom.tex; April 12, 2006; 17:59; p. 31 ...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Vectors

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