presentacion de variedades
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Control &Dynamical Systems
CA L T EC H
Manifolds, Mappings, Vector Fields
Jerrold E. Marsden
Control and Dynamical Systems, Caltech
http://www.cds.caltech.edu/ marsden/
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Some History
Carl Friedrich Gauss
Born: 30 April 1777 in Brunswick (Germany)
Died: 23 Feb 1855 in Gottingen (Germany)
PhD, 1799, University of Helmstedt, advisor was Pfaff. Dissertation
was on the fundamental theorem of algebra. 1801, predicted position of asteroid Ceres; used averaging methods;
famous as an astronomer; 1807, director of the Gottingen observatory
18011830, Continued to work on algebra, and increasingly in geometry
of surfaces, motivated by non-Euclidean geometry. 1818, Fame as a geologist; used geometry to do a geodesic survey.
1831, work on potential theory (Gauss theorem), terrestrial mag-netism, etc.
1845 to 1851, worked in finance and got rich.2
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Some History
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Some History
Carl Gustav Jacob Jacobi
Born: 10 Dec 1804 in Potsdam
Died: 18 Feb 1851 in Berlin
1821, entered the University of Berlin
1825, PhD Disquisitiones Analyticae de Fractionibus Simplicibus
1825-26 Humboldt-Universitat, Berlin
18261844 University of Konigsberg
1827, Fundamental work on elliptic functions 18441851 University of Berlin
Lectures on analytical mechanics (Berlin, 1847 - 1848); gave a de-tailed and critical discussion of Lagranges mechanics.
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Some History
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Some History
Georg Friedrich Bernhard Riemann
Born: 17 Sept 1826 in Breselenz (Germany)
Died: 20 July 1866 in Selasca, Italy
PhD, 1851, Gottingen, advisor was Gauss; thesis in complex analysis
on Riemann surfaces 1854, Habilitation lecture: Uber die Hypothesen welche der Geome-
trie zu Grunde liegen (On the hypotheses that lie at the foun-dations of geometry), laid the foundations of manifold theory and
Riemannian geometry 1857, Professor at Gottingen
1859, Reported on the Riemann hypothesis as part of his inductioninto the Berlin academy
1866, Died at the age of 40 in Italy,6
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Some History
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Some History
Felix Klein
Born: 25 April 1849 in Dsseldorf, Germany
Died: 22 June 1925 in Gottingen, Germany
1871; University of Christiania (the city which became Kristiania, thenOslo in 1925)
PhD, 1868, University of Bonn under Plucker, on applications of ge-ometry to mechanics.
1872, professor at Erlangen, in Bavaria; laid foundations of geometryand how it connected to group theory (Erlangen Programm); collabo-rations with Lie.
1880 to 1886, chair of geometry at Leipzig
1886, moved to Gottingen to join Hilberts group
18901900+, More on mathematical physics, mechanics; correspondedwith Poincare8
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Some History
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Some History
Marius Sophus Lie
Born: 17 Dec 1842 in Nordfjordeide, Norway
Died: 18 Feb 1899 in Kristiania (now Oslo), Norway
1871; University of Christiania (the city which became Kristiania, then
Oslo in 1925) PhD, 1872, University of Christiania On a class of geometric trans-
formations
18861898 University of Leipzig (got the chair of Klein); where he wrote
his famous 3 volume work Theorie der Transformationsgruppen 1894, Kummer was his student at Leipzig
18981899, Returned to the University of Christiania; dies shortly afterhis return.
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Some History
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Some History
Henri Poincare
Born: 29 April 1854 in Nancy, Lorraine, France
Died: 17 July 1912 in Paris, France
1879, Ph.D. University of Paris, Advisor: Charles Hermite
1879, University of Caen 1881, Faculty of Science in Paris in 1881
1886, Sorbonne and Ecole Polytechnique until his death at age 58
1887, elected to the Acadmie des Sciences; 1906, President.
1895, Analysis situs(algebraic topology)
18921899, Les Methodes nouvelles de la mechanique celeste (3 vol-umes) and, in 1905, Lecons de mecanique celeste
His cousin, Raymond Poincare, was President of France during WorldWar I.
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Manifolds
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Manifolds
Manifolds are sets M that locally look like linear
spacessuch as Rn.
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Manifolds
Manifolds are sets M that locally look like linear
spacessuch as Rn.
A chart on M is a subset U of M together with abijective map : U (U) Rn.
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Manifolds
Manifolds are sets M that locally look like linear
spacessuch as Rn.
A chart on M is a subset U of M together with abijective map : U (U) Rn.
Usually, we denote (m) by (x1, . . . , xn) and call thexi the coordinates of the point m U M.
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Manifolds
Manifolds are sets M that locally look like linear
spacessuch as Rn.A chart on M is a subset U of M together with a
bijective map : U (U) Rn.
Usually, we denote (m) by (x1, . . . , xn) and call thexi the coordinates of the point m U M.
Two charts (U,) and (U, ) such that U U =
are called compatible if(U U
) and
(U
U)are open subsets ofRn and the maps
1|(U U) : (U U) (U U)
and
()1|(U U) : (U U) (U U)14
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Manifolds
are C. Here, 1|(UU) denotes the restric-
tion of the map 1 to the set (U U).
(U)
(U)
U
M
x1
x1
xn
x
n
xn
U
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Manifolds
We call M a differentiable n-manifold when:
M1. The setM is covered by a collection of charts, that is, everypoint is represented in at least one chart.
M2. M has an atlas; that is, M can be written as a union ofcompatible charts.
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f
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Manifolds
We call M a differentiable n-manifold when:
M1. The setM is covered by a collection of charts, that is, everypoint is represented in at least one chart.
M2. M has an atlas; that is, M can be written as a union ofcompatible charts.
Differentiable structure: Start with a given atlasand (be democratic and) include all charts compatiblewith the given ones.
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M if ld
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Manifolds
We call M a differentiable n-manifold when:
M1. The setM is covered by a collection of charts, that is, everypoint is represented in at least one chart.
M2. M has an atlas; that is, M can be written as a union ofcompatible charts.
Differentiable structure: Start with a given atlasand (be democratic and) include all charts compatiblewith the given ones.
Example: Start with R3 as a manifold with simplyone (identity) chart. We then allow other charts suchas those defined by spherical coordinatesthis is then adifferentiable structure on R3. This will be understoodto have been done when we say we have a manifold.
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M if ld
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Manifolds
Topology. Every manifold has a topology obtained
by declaring open neighborhoods in charts to be openneighborhoods when mapped to M by the chart.
Tangent Vectors. Two curves t c1(t) and t
c2(t) in an n-manifold Mare called equivalent at thepoint m if
c1(0) = c2(0) = m
andddt
( c1)t=0
= ddt
( c2)t=0
in some chart .
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M if ld
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Manifolds
0
I
c1
c2
m
F
U
R
U
M
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T V
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Tangent Vectors
A tangent vector v to a manifold M at a point
m M is an equivalence class of curves at m.
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T V
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Tangent Vectors
A tangent vector v to a manifold M at a point
m M is an equivalence class of curves at m. Set of tangent vectors to M at m is a vector space.
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T V
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Tangent Vectors
A tangent vector v to a manifold M at a point
m M is an equivalence class of curves at m. Set of tangent vectors to M at m is a vector space.
Notation: TmM = tangent space to M at m M.
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T t V t
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Tangent Vectors
A tangent vector v to a manifold M at a point
m M is an equivalence class of curves at m. Set of tangent vectors to M at m is a vector space.
Notation: TmM = tangent space to M at m M.
We think of v TmM as tangent to a curve in M.
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T t V t
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Tangent Vectors
m
TmM
v
M
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M
gives associated coordinates (x1, . . . , xn) for points in U.
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M gives associated coordinates (x1, . . . , xn) for points in U.
Let v TmM be a tangent vector to M at m;
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M gives associated coordinates (x1, . . . , xn) for points in U.
Let v TmM be a tangent vector to M at m;
c a curve representative of the equivalence class v.
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M gives associated coordinates (x1, . . . , xn) for points in U.
Let v TmM be a tangent vector to M at m;
c a curve representative of the equivalence class v.
The components ofv are the numbers v1, . . . , vn defined by tak-ing the derivatives of the components of the curve c:
vi =d
dt( c)i
t=0
,
where i = 1, . . . , n.
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T t V t
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Tangent Vectors
Components of a tangent vector.
: U M Rn a chart M gives associated coordinates (x1, . . . , xn) for points in U.
Let v TmM be a tangent vector to M at m;
c a curve representative of the equivalence class v.
The components ofv are the numbers v1, . . . , vn defined by tak-ing the derivatives of the components of the curve c:
vi =d
dt( c)i
t=0
,
where i = 1, . . . , n. Components are independent of the representative curve chosen, but
they do depend on the chart chosen.
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Ta e t B dles
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Tangent Bundles
Tangent bundle ofM, denoted by TM, is the dis-
joint union of the tangent spaces to M at the pointsm M, that is,
TM = mM
TmM.
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Tangent Bundles
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Tangent Bundles
Tangent bundle ofM, denoted by TM, is the dis-
joint union of the tangent spaces to M at the pointsm M, that is,
TM = mM
TmM.
Points ofTM: vectors v tangent at some m M.
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Tangent Bundles
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Tangent Bundles
Tangent bundle ofM, denoted by TM, is the dis-
joint union of the tangent spaces to M at the pointsm M, that is,
TM =
mM
TmM.
Points ofTM: vectors v tangent at some m M.
IfM is an n-manifold, then TM is a 2n-manifold.
The natural projection is the map M
: TM Mthat takes a tangent vector v to the point m M atwhich the vector v is attached.
The inverse image 1M (m) of m M is the tangent
space TmMthe fiber ofTM over the point m M.22
Differentiable Maps
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Differentiable Maps
A map f : M N is differentiable (resp. Ck) if
in local coordinates on M and N, the map f is repre-sented by differentiable (resp. Ck) functions.
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Differentiable Maps
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Differentiable Maps
A map f : M N is differentiable (resp. Ck) if
in local coordinates on M and N, the map f is repre-sented by differentiable (resp. Ck) functions.
The derivative off : M N at a point m M is
a linear map Tmf : TmM Tf(m)N. Heres how: For v TmM, choose a curve c in M with c(0) = m, and velocity
vector dc/dt |t=0 = v .
Tmf v is the velocity vector at t = 0 of the curve f c : R N,
that is,Tmf v =
d
dtf(c(t))
t=0
.
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Chain Rule
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Chain Rule
The vector Tmfv does not depend on the curve c but
only on the vector v.
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Diffeomorphisms
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Diffeomorphisms
A differentiable (or of class Ck) map f : M N
is called a diffeomorphism if it is bijective and itsinverse is also differentiable (or of class Ck).
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Diffeomorphisms
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Diffeomorphisms
A differentiable (or of class Ck) map f : M N
is called a diffeomorphism if it is bijective and itsinverse is also differentiable (or of class Ck).
IfTmf : TmM Tf(m)N is an isomorphism, the in-
verse function theorem states that f is a localdiffeomorphism around m M
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Diffeomorphisms
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Diffeomorphisms
A differentiable (or of class Ck) map f : M N
is called a diffeomorphism if it is bijective and itsinverse is also differentiable (or of class Ck).
IfTmf : TmM Tf(m)N is an isomorphism, the in-
verse function theorem states that f is a localdiffeomorphism around m M
The set of all diffeomorphisms f : M M formsa group under composition, and the chain rule showsthat T(f1) = (Tf)1.
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Submanifolds
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Submanifolds
Iff : M N is a smooth map, a point m M is a
regular point ifTmf is surjective; otherwise, m is acritical point off. A value n N is regular if allpoints mapping to n are regular.
Submersion theorem: Iff : M N is a smoothmap and n is a regular value of f, then f1(n) is asmooth submanifold ofMof dimension dim Mdim Nand
Tmf1(n)
= ker Tmf.
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Vector Fields and Flows
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Vector Fields and Flows
Vector field X on M: a map X : M TM that
assigns a vector X(m) at the point m M; that is,M X= identity.
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Vector Fields and Flows
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Vector Fields and Flows
Vector field X on M: a map X : M TM that
assigns a vector X(m) at the point m M; that is,M X= identity.
Vector space of vector fields on M is denoted X(M).
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Vector Fields and Flows
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Vector Fields and Flows
Vector field X on M: a map X : M TM that
assigns a vector X(m) at the point m M; that is,M X= identity.
Vector space of vector fields on M is denoted X(M).
An integral curve ofXwith initial condition m0 att = 0 is a (differentiable) map c : ]a, b[ M such that]a, b[ is an open interval containing 0, c(0) = m0, and
c
(t) = X(c(t))for all t ]a, b[; i.e., a solution curve of this ODE.
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Vector Fields and Flows
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Vector Fields and Flows
Vector field X on M: a map X : M TM that
assigns a vector X(m) at the point m M; that is,M X= identity.
Vector space of vector fields on M is denoted X(M).
An integral curve ofXwith initial condition m0 att = 0 is a (differentiable) map c : ]a, b[ M such that]a, b[ is an open interval containing 0, c(0) = m0, and
c(t) = X(c(t))
for all t ]a, b[; i.e., a solution curve of this ODE.
Flow of X: The collection of maps t : M Msuch that t t(m) is the integral curve ofX with
initial condition m.28
Vector Fields and Flows
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Vector Fields and Flows
Existence and uniqueness theorems from ODE guaran-
tee that exists is smooth in m and t (where defined)ifX is.
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Vector Fields and Flows
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Vector Fields and Flows
Existence and uniqueness theorems from ODE guaran-
tee that exists is smooth in m and t (where defined)ifX is.
Uniqueness flow property
t+s = t s
along with the initial condition 0 = identity.
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Vector Fields and Flows
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Vector Fields and Flows
An important fact is that ifX is a Cr vector field, then
its flow is a Cr map in all variables.
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Vector Fields and Flows
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Vector Fields and Flows
An important fact is that ifX is a Cr vector field, then
its flow is a Cr map in all variables.An important estimate that is the key to proving things
like this is the Gronwall inequality: if on an inter-
val [a, b], we havef(t) A +
ta
f(s)g(s)ds
where A 0, and f, g 0 are continuous, then
f(t) A exp
t
a
g(s)ds
This is used on the integral equation defining a flow
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Vector Fields and Flows
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Vector Fields and Flows
Strategy for proving that the flow is C1:
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Vector Fields and Flows
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Vector Fields and Flows
Strategy for proving that the flow is C1:
Formally linearize the equations and find the equationthat Dt should satisfy. This is the first variationequation.
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Vector Fields and Flows
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Vector Fields and Flows
Strategy for proving that the flow is C1:
Formally linearize the equations and find the equationthat Dt should satisfy. This is the first variationequation.
Use the basic existence theory to see that the first vari-ation equation has a solution.
Show that this solution actually satisfies the definition
of the derivative oft by making use of Gronwall.
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Vector Fields and Flows
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Notion ofcompleteness of vector fields ; integral
curves are defined for all time.Criteria discussed in book; an important case is that
if, for example, by making a priori estimates, one can
show that any integral curve defined on a finite opentime interval (a, b) remains in a compact set, then thevector field is complete.
For instance sometimes such an estimate can be ob-
tained as an energy estimate; eg, the flow of x +x3 = 0 is complete since it has the preserved energyH(x, x) = 12x
2 + 14x4, which confines integral curves to
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Differential of a Function
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Derivative of f : M R at m M: gives a map
Tmf : TmM Tf(m)R = R.
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Differential of a Function
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Derivative of f : M R at m M: gives a map
Tmf : TmM Tf(m)R = R.Gives a linear map df(m) : TmM R.
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Differential of a Function
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Derivative of f : M R at m M: gives a map
Tmf : TmM Tf(m)R = R.Gives a linear map df(m) : TmM R.
Thus, df(m) TmM, the dual ofTmM.
If we replace each vector space TmM with its dualTmM, we obtain a new 2n-manifold called the cotan-gent bundle and denoted by TM.
We call df the differential off. For v TmM, wecall df(m) v the directional derivative of f inthe direction v.
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Differential of a Function
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In a coordinate chart or in linear spaces, this notion co-
incides with the usual notion of a directional derivativelearned in vector calculus.
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Vector Fields as Differential Operators
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Specifying the directional derivatives completely deter-
mines a vector, and so we can identify a basis ofTmMusing the operators /xi. We write
{e1, . . . , en} =
x1, . . . ,
xn
for this basis, so that v = vi/xi.
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Vector Fields as Differential Operators
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The dual basis to /xi is denoted by dxi. Thus, rel-
ative to a choice of local coordinates we get the basicformula
df(x) =f
xidxi
for any smooth function f : M R.
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Vector Fields as Differential Operators
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The dual basis to /xi is denoted by dxi. Thus, rel-
ative to a choice of local coordinates we get the basicformula
df(x) =f
xidxi
for any smooth function f : M R.We also have
X[f] = Xif
xi
which is why we write
X= Xi
xi
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Jacobi-Lie Bracket
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Given two vector fields X and Y, here is a unique
vector field [X, Y] such that as a differential operator,[X, Y][f] = X[Y[f]] Y[X[f]]
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Jacobi-Lie Bracket
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Given two vector fields X and Y, here is a unique
vector field [X, Y] such that as a differential operator,[X, Y][f] = X[Y[f]] Y[X[f]]
Easiest to see what is going on by computing the com-
mutator in coordinates:
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Jacobi-Lie Bracket
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We get
[X, Y] =Xi
xi, Yj
xj
=
Xi
xiYj
xj Yj
xjXi
xi
= XiYj
xi
xj Yj
Xi
xj
xi
= XiYj
xi Yi
Xj
xi
xj
Note that the second derivative terms 2/xixj can-celed out. This is the secret why the bracket is a