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\end_layout

\begin_layout Title
A short introduction to differential geometry
\begin_inset Newline newline
\end_inset

from the perspective of tensor calculus
\end_layout

\begin_layout Author
Christopher Albert
\end_layout

\begin_layout Address
Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching,
 Germany
\begin_inset Newline newline
\end_inset

<albert@alumni.tugraz.at>
\end_layout

\begin_layout Abstract
In this text a short explanation on the treatment of space as a smooth manifold
 and phase-space as a symplectic manifold for readers acquainted to terminology
 from tensor calculus in co-/contravariant notation is given.
 The main aim of this text is to provide a fast track to working knowledge
 of the discussed concepts.
 For more details, the three textbooks of Arnold, Jose/Saletan and Marsden
 should be helpful for further clarifications and deeper understanding.
 For a thorough introduction into smooth manifolds, the video lectures of
 Frederic Schuller from the WE-Heraeus International Winter School on Gravity
 and Light are highly recommended.
 A review concerned with guiding-center dynamics and in more traditional
 notation is given by Cary/Brizard at 
\begin_inset CommandInset href
LatexCommand href
target "https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.81.693"

\end_inset

 .
 A main feature distinguishing differential geometry from tensor calculus
 is the treatment of covectors as first-class objects in addition to vectors.
 As a concequence in many derivations no metric tensor has to be specified,
 and the construction of structure-preserving numerical methods becomes
 simpler and more transparent.
 Apart from conceptual differences, the practical use of the formalisms
 is similar, in particular the notation with upper and lower indexes for
 contravariant and covariant components and the convention to sum over indexes
 appearing once up and once down.
\end_layout

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\begin_inset ERT
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makeatother
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\end_layout

\begin_layout Section
Notation
\end_layout

\begin_layout Standard
Here we fix some notational conventions for the following text.
 Scalars and scalar functions are denoted in lowercase italic letters 
\begin_inset Formula $a,f$
\end_inset

 and for points 
\begin_inset Formula $X,Q,Z$
\end_inset

 we use uppercase italic letters.
 We use italic bold letters to denote ensor fields, in particular vectors
 
\begin_inset Formula $\dot{\v q},\dot{\v z}$
\end_inset

 and covectors 
\begin_inset Formula $\v p,\v A$
\end_inset

.
 Coordinate tuples are denoted by bold upright letters 
\begin_inset Formula $\xset,\zset,\qset,\pset$
\end_inset

.
 Spatial coordinates 
\begin_inset Formula $\xset$
\end_inset

 have components 
\begin_inset Formula $x^{k}$
\end_inset

, canonical coordinates 
\begin_inset Formula $\qset,\pset$
\end_inset

 consist of tuples 
\begin_inset Formula $q^{i},p_{j}$
\end_inset

, and generally non-canonical coordinates in phase-space 
\begin_inset Formula $\zset$
\end_inset

 have entries 
\begin_inset Formula $z^{\alpha}$
\end_inset

.
 Lower index notation for canonical momenta 
\begin_inset Formula $p_{j}$
\end_inset

 is used since in the classical construction 
\begin_inset Formula $p_{j}$
\end_inset

 are not only coordinates in phase-space but also components of covectors
 with respect to configuration space charted by coordinates 
\begin_inset Formula $q^{i}$
\end_inset

.
 Latin indices 
\begin_inset Formula $i,j,k,l$
\end_inset

 run from 
\begin_inset Formula $1$
\end_inset

 to 
\begin_inset Formula $3$
\end_inset

 for spatial coordinates 
\begin_inset Formula $x^{k}$
\end_inset

, or from 
\begin_inset Formula $1$
\end_inset

 to 
\begin_inset Formula $N$
\end_inset

 for canonical coordinates in a system with 
\begin_inset Formula $N$
\end_inset

 degrees of freedom.
 Greek indices like 
\begin_inset Formula $\alpha,\beta$
\end_inset

 are used for general phase-space coordinates 
\begin_inset Formula $z^{\alpha}$
\end_inset

 and run from 
\begin_inset Formula $1$
\end_inset

 to 
\begin_inset Formula $2N$
\end_inset

.
 We use the usual convention from tensor algebra to sum over indices appearing
 once up and once down in formulas, i.e.
 
\begin_inset Formula $p_{i}\dot{q}^{i}\equiv\sum_{i}p_{i}\dot{q}^{i}$
\end_inset

, where indexes in the denominator of derivatives switch their position,
 e.g.
 
\begin_inset Formula $\partial H/\partial q^{j}=p_{j}$
\end_inset

 has a lower index.
 A common convention of physics literature is used where functional dependencies
 such as 
\begin_inset Formula $\zset(\qset,\pset),\zset(t)$
\end_inset

 are denoted by the same letters as 
\begin_inset Formula $\zset$
\end_inset

 alone.
 If arguments are not written explicitly inside functions the meaning should
 become clear from the respective context.
\end_layout

\begin_layout Section
Basics
\end_layout

\begin_layout Standard
Physics can be treated on different levels of abstraction.
 Here we will start from the simplest picture and progress to the most advanced
 one, expressing our dissatisfaction with the result in each step, to justify
 an increase in the level of abstraction.
 The names of the sub-sections are chosen from the author's experience and
 can of course vary from school to school.
 The last subsection on the mathematical theory of differential geometry
 is at the highest level of abstraction and clarity and contains all other
 section as special cases and/or more sloppy interpretations.
 This does not necesserily mean that one should start on the top.
 In the view of the author it is rather convenient to start from the most
 basic levels and see where conceptual problems arise.
 Once the highest level is reached, one can take a step back to the sometimes
 more practical levels, with the power of understanding what objects we
 are actually talking about, and how they are interrelated.
\end_layout

\begin_layout Subsection
High-school physics
\end_layout

\begin_layout Standard
The usual picture of high-school mathematics and physics is the following:
 The world of classical physics can be described by points 
\begin_inset Formula $X$
\end_inset

 given in Cartesian coordinates 
\begin_inset Formula $\xset=(x,y,z)$
\end_inset

, and time 
\begin_inset Formula $t$
\end_inset

 that is running steadily.
 A vector is constructed as an arrow pointing from 
\begin_inset Formula $X_{1}$
\end_inset

 to 
\begin_inset Formula $X_{2}$
\end_inset

 with
\begin_inset Formula 
\begin{equation}
\v u_{12}=\left(\begin{array}{c}
u_{12\,x}\\
u_{12\,y}\\
u_{12\,z}
\end{array}\right)\equiv\left(\begin{array}{c}
x_{2}-x_{1}\\
y_{2}-y_{1}\\
z_{2}-z_{1}
\end{array}\right).
\end{equation}

\end_inset

It is said to exists independently from its spreading point 
\begin_inset Formula $P$
\end_inset

 and can be parallely translated in space.
 The length of a vector can be computed by its norm
\begin_inset Formula 
\begin{equation}
|\v u|=\sqrt{u_{x}^{\,2}+u_{y}^{\,2}+u_{z}^{\,2}}.
\end{equation}

\end_inset

To each point 
\begin_inset Formula $P$
\end_inset

 we can assign a vector from the origin point 
\begin_inset Quotes eld
\end_inset


\begin_inset Formula $0$
\end_inset


\begin_inset Quotes erd
\end_inset

, 
\begin_inset Formula 
\begin{equation}
\v r(P)\equiv\v u^{0P}=\left(\begin{array}{c}
x\\
y\\
z
\end{array}\right),
\end{equation}

\end_inset

whose components are the same as the coordinates of 
\begin_inset Formula $P$
\end_inset

.
 The inner product between vectors 
\begin_inset Formula $\v u$
\end_inset

 and 
\begin_inset Formula $\v v$
\end_inset

 is given by
\begin_inset Formula 
\begin{equation}
\v u\cdot\v v=u_{x}v_{x}+u_{y}v_{y}+u_{z}v_{z}.
\end{equation}

\end_inset

By geometrical means we see that can be related to the angle 
\begin_inset Formula $\theta^{\v{uv}}$
\end_inset

 between the vectors via
\begin_inset Formula 
\begin{equation}
\v u\cdot\v v=|\v u||\v v|\cos\theta^{\v u\v v}.
\end{equation}

\end_inset

We also notice that the norm is related to the inner product via 
\begin_inset Formula $|\v u|=\sqrt{\v u\cdot\v u}$
\end_inset

.
 In addition we introduce the cross product
\begin_inset Formula 
\begin{equation}
\v u\times\v v\equiv\left(\begin{array}{c}
u_{y}u_{z}-u_{z}u_{y}\\
u_{z}u_{x}-u_{x}u_{z}\\
u_{x}u_{y}-u_{y}u_{x}
\end{array}\right),
\end{equation}

\end_inset

yielding a vector as a result.
\end_layout

\begin_layout Subsubsection*
Mechanics
\end_layout

\begin_layout Standard
A particle that starts at time 
\begin_inset Formula $t=0$
\end_inset

 at point 
\begin_inset Formula $X$
\end_inset

 with a constant velocity vector 
\begin_inset Formula $\v v$
\end_inset

 will have a time-dependent position vector
\begin_inset Formula 
\begin{equation}
\v r(t)=\v r(0)+\v vt
\end{equation}

\end_inset

that we can then translate back to a point 
\begin_inset Formula $X(t)$
\end_inset

.
 If a constant vectorial force 
\begin_inset Formula $\v F$
\end_inset

 acts on a particle of mass 
\begin_inset Formula $m$
\end_inset

 we obtain an accelaration vector 
\begin_inset Formula 
\begin{equation}
\v a\equiv\frac{\d\v v(t)}{\d t}=\frac{\v F}{m}
\end{equation}

\end_inset

according to Newton's law.
 We then first compute the time-dependent velocity vector by integrating
 once,
\begin_inset Formula 
\begin{equation}
\v v(t)=\v v(0)+\frac{\v F}{m},
\end{equation}

\end_inset

and integrate another time to obtain
\begin_inset Formula 
\begin{equation}
\v r(t)=\v r(0)+\v v(0)t+\frac{\v F}{2m}t^{2}.
\end{equation}

\end_inset


\end_layout

\begin_layout Subsubsection*
Field theory
\end_layout

\begin_layout Standard
A vector fields 
\begin_inset Formula $\v E(X)$
\end_inset

 means to attach a vector to each point 
\begin_inset Formula $X$
\end_inset

 in space, i.e.
 
\begin_inset Formula $\v E$
\end_inset

 is like a function on components 
\begin_inset Formula $\xset=(x,y,z)$
\end_inset

 that has vectors as an output.
 In that sense we can view 
\begin_inset Formula $\v r(X)$
\end_inset

 as a vector field that yields Cartesian components of 
\begin_inset Formula $X$
\end_inset

 in each point.
 For electromagnetism and gravity, the intuitive picture of a 
\emph on
flow
\emph default
 of a vector field through a surface is introduced.
 There we take the normal components of a vector field to the surface, and
 sum them up.
 Gauss' law can be derived by looking at the vectors in the direction of
 a point charge and how they flow through a spherical surface.
\end_layout

\begin_layout Subsubsection*
Why we are dissatisfied
\end_layout

\begin_layout Standard
First of all, we could only use Cartesian coordinates.
 We practically identified points, vectors, and their Cartesian components,
 despite noticing that the inner product would work the same, if we translated
 or rotated our coordinate frame.
 In mechanics and field theory we were limited to constant forces and simple
 fields.
 For treating cylindrical and spherical symmetry we had to rely on ad-hoc
 solutions by drawing arrows, since using Cartesian coordinates there is
 very cumbersome.
 The questions what space and time actually are, and why we can describe
 them by numbers in the given way, remains completely open.
\end_layout

\begin_layout Subsection
Undergraduate physics
\end_layout

\begin_layout Standard
In the undergraduate physics courses we keep the picture of vector calculus
 and classical physics from high-school, but more advanced concepts are
 introduced on several levels.
 A vector is generalized from a geometric quantity to an abstract object
 in a linear vector space that allows for addition and multiplication by
 scalars.
 Newton's equations of motion are generalized to Lagrange's and Hamilton's
 equations in generalized coordinates, and solved for space- and time-dependent
 forces.
 In field theory we solve partial differential equations in possibly curvilinear
 coordinates.
 Such a coordinate change is given by a set of transformation equations
\begin_inset Formula 
\begin{align}
x & =x(\xset)\equiv x(x^{1},x^{2},x^{3}),\\
y & =y(\xset)\equiv y(x^{1},x^{2},x^{3}),\\
z & =z(\xset)\equiv z(x^{1},x^{2},x^{3}),
\end{align}

\end_inset

or more compactly 
\begin_inset Formula $\v r=\v r(\xset)$
\end_inset

, with the position vector 
\begin_inset Formula $\v r$
\end_inset

 still identified with its Cartesian coordinates 
\begin_inset Formula $(x,y,z)$
\end_inset

 via
\begin_inset Formula 
\begin{align}
\v r & \equiv x\hat{\v e}_{x}+y\hat{\v e}_{y}+z\hat{\v e}_{z}
\end{align}

\end_inset

even though we already anticipate that the concept of a position 
\begin_inset Quotes eld
\end_inset

vector
\begin_inset Quotes erd
\end_inset

 becomes problematic due to the possibility of coordinate changes.
 If we want to be more precise and distinguish 
\begin_inset Formula $X$
\end_inset

 from 
\begin_inset Formula $\v r$
\end_inset

, we look at the position vector field 
\begin_inset Formula $\v r(X)$
\end_inset

 and represent 
\begin_inset Formula $X(\xset)$
\end_inset

 via coordinates 
\begin_inset Formula $\xset$
\end_inset

, so when we write 
\begin_inset Formula $\v r$
\end_inset

, we actually mean the vector-valued function 
\begin_inset Formula $\v r(\xset)\equiv\v r(X(\xset))$
\end_inset

 representing the vector field 
\begin_inset Formula $\v r(X)$
\end_inset

 in coordinates 
\begin_inset Formula $\xset$
\end_inset

.
 Unit vectors in curvilinear coordinates are given as derivatives of this
 function via
\begin_inset Formula 
\begin{equation}
\hat{\v e}_{k}=\frac{1}{|\partial\v r/\partial x^{k}|}\frac{\partial\v r}{\partial x^{k}},
\end{equation}

\end_inset

where we require an orthonormal set, i.e.
 
\begin_inset Formula $\hat{\v e}_{k}\cdot\hat{\v e}_{l}=\delta_{kl}$
\end_inset

, so 
\begin_inset Formula $0$
\end_inset

 if 
\begin_inset Formula $k\neq l$
\end_inset

 and 
\begin_inset Formula $1$
\end_inset

 if 
\begin_inset Formula $k=l$
\end_inset

.
 This is the case for many sets of coordinates, including cylindrical and
 spherical coordinates, that make our life much easier in electromagnetic
 field theory.
 To manipulate vector fields we introduce the nabla symbol 
\begin_inset Formula $\nabla$
\end_inset

 as a vector differential operator, producing gradient, curl, and divergence
 via
\begin_inset Formula 
\begin{align}
\nabla\Phi & \equiv\frac{\partial\Phi}{\partial x}\v e_{x}+\frac{\partial\Phi}{\partial y}\v e_{y}+\frac{\partial\Phi}{\partial z}\v e_{z},\\
\nabla\times\v A & \equiv\left(\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z}\right)\v e_{x}+\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x}\right)\v e_{y}+\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{y}}{\partial x}\right)\v e_{z},\\
\nabla\cdot\v B & \equiv\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial y}+\frac{\partial B_{z}}{\partial z}.
\end{align}

\end_inset

We proof differential theorems of vector calculus, namely
\begin_inset Formula 
\begin{equation}
\nabla\times(\nabla\Phi)=0,\quad\nabla\cdot(\nabla\times\v A)=0,
\end{equation}

\end_inset

that aid us in the construction of potential fields in electrodynamics.
 Related integral theorems of Gauss and Stokes are
\begin_inset Formula 
\begin{equation}
\int_{\Omega_{3}}\nabla\cdot\v D\,\d V=\int_{\partial\Omega_{3}}\v D\cdot\d\v S,\quad\int_{\Omega_{2}}(\nabla\times\v D)\cdot\d\v S=\int_{\partial\Omega_{2}}\v D\cdot\d\v l,
\end{equation}

\end_inset

where 
\begin_inset Formula $\Omega_{3}$
\end_inset

 and 
\begin_inset Formula $\Omega_{2}$
\end_inset

 are 3- and 2-dimensional domains, and 
\begin_inset Formula $\Omega_{3},\Omega_{2}$
\end_inset

 their respective boundaries with surface normal 
\begin_inset Formula $\d\v S$
\end_inset

 and line element 
\begin_inset Formula $\d\v l$
\end_inset

 being infinitesimal vectors.
\end_layout

\begin_layout Subsubsection*
Why we are dissatisfied
\end_layout

\begin_layout Standard
Transforming differential operators to curvilinear coordinates is complicated
 and we are limited to normalized basis vectors in orthogonal coordinate
 systems.
\end_layout

\begin_layout Subsection
Graduate level physics
\end_layout

\begin_layout Standard
Still the focus of tensor calculus in curvilinear geometry is on the components
 of a tensor in a certain basis.
 This is reflected in the statement that a tensor is a 
\begin_inset Quotes eld
\end_inset

matrix with transformation properties under coordinate changes
\begin_inset Quotes erd
\end_inset

.
\end_layout

\begin_layout Subsubsection*
Why we are dissatisfied
\end_layout

\begin_layout Standard
The fundamental building block of the co-contravariant treatment in classical
 physics is still based on a position vector field 
\begin_inset Formula $\v r(x,y,z)$
\end_inset

 given in Cartesian coordinates.
 In general relativity the absence of such a vector field confuses matter
 further, since tensors are now only given in terms of their components
 fulfilling transformation properties 
\begin_inset Quotes eld
\end_inset

out of thin air
\begin_inset Quotes erd
\end_inset

, without a global reference frame.
 While we know what quantities are better represented in co- or contravariant
 form, there is no clear distinction between the two.
\end_layout

\begin_layout Subsection
Differential geometry
\end_layout

\begin_layout Standard
One of the features of a mathematical description is the exact specification
 of what objects we are dealing with, and how to trace them back to already
 known mathematical objects, going down to fundamental axioms that are chosen
 to make intuitive concepts work, while giving them a solid foundation.
 Similarly the connection to the physical world relies on postulates.
 In contrast to axioms they can be justified by the correctness of predictions
 made by the theory constructed in terms of these objects.
 As common in mathematical descriptions, differential geometry strives to
 be economic with additional structure, such as metrizability or differentiabili
ty, requiring only the bare minimum for the according task.
 On first sight this looks like this means departing from reality and applicatio
n, but the contrary is the case.
 The mathematical approach is more honest in the sense that we do not claim
 to know what our system actually is, but rather distill out of it the bare
 minimum properties we require it to have in order to make sense.
 This allows us to separate topological from metric features and gives a
 clear guideline on structure-preserving discretizations such as symplectic
 time-stepping schemes and vector finite elements.
 Here we go back to treating vectors in their geometrical sense rather than
 abstract vectors, and give a more precise definition to what that means.
\end_layout

\begin_layout Subsubsection*
Space
\end_layout

\begin_layout Standard
Here we describe the mathematical foundation to describe physical space,
 which applies also to spacetime and phase-space with slight modifications.
 In the philosophy of the bare minimum we start with modeling space as a
 
\emph on
set
\emph default
 
\begin_inset Formula $\mathbb{M}$
\end_inset

 of points 
\begin_inset Formula $X\in\mathbb{M}$
\end_inset

.
 Our goal is to add a sufficient amount of mathematical structure to this
 set to be able to describe motion and fields in different sets of coordinates
 and formulate dynamical laws as differential equations.
\end_layout

\begin_layout Enumerate
Since we would like to know what points are adjacent to each other, we require
 a topology 
\begin_inset Formula $\mathcal{O}$
\end_inset

 that yields the notion of 
\emph on
open neighbourhoods
\emph default
 containing adjacent points.
 This introduces the concept of 
\emph on
continuous maps
\emph default
 between topological spaces that keep neighbouring points adjacent to each
 other also in their image.
 Adding this structure to 
\begin_inset Formula $\mathbb{M}$
\end_inset

 makes it a 
\emph on
topological space 
\begin_inset Formula $(\mathbb{M},\mathcal{O})$
\end_inset

.
\end_layout

\begin_layout Enumerate
In order to be able to translate points to numbers, we require 
\begin_inset Formula $\mathbb{M}$
\end_inset

 to be at least locally similar to Euclidian space 
\begin_inset Formula $\mathbb{R}^{d}$
\end_inset

 of dimension 
\begin_inset Formula $d$
\end_inset

 equiped with the usual open sets as a topology based on the Euclidian distance.
 This means that for each open set of points 
\begin_inset Formula $\Omega\subset\mathbb{M}$
\end_inset

 we can find a continuous and invertible map to 
\begin_inset Formula $\mathbb{R}^{d}$
\end_inset

 which yields points as functions 
\begin_inset Formula $X(\xset)$
\end_inset

 of 
\begin_inset Formula $d$
\end_inset

 coordinates 
\begin_inset Formula $\xset=(x^{1},\dots,x^{d})$
\end_inset

.
 Such a map is called a 
\emph on
chart.

\emph default
 A collection of charts is called an 
\emph on
atlas 
\emph default

\begin_inset Formula $\mathcal{A}$
\end_inset

 that contains at least one chart for each 
\begin_inset Formula $\Omega$
\end_inset

.
 With 
\begin_inset Formula $d$
\end_inset

 the minimum required chart dimension
\begin_inset Foot
status open

\begin_layout Plain Layout
We cannot chart the surface of the earth in 
\begin_inset Formula $\mathbb{R}^{1}$
\end_inset

, and 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 would be too much, so we use 
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset


\end_layout

\end_inset

 we call 
\begin_inset Formula $(\mathbb{M},\mathcal{O},\mathcal{A})$
\end_inset

 a 
\begin_inset Formula $d$
\end_inset


\emph on
-dimensional topological manifold
\emph default
.
\end_layout

\begin_layout Enumerate
To define what differentiation means on the manifold, we require the 
\emph on
transition maps
\emph default
 representing coordinate transformations between regions where charts overlap
 to be differentiable in 
\begin_inset Formula $\mathbb{R}^{d}$
\end_inset

.
 It can be proven that already requiring 
\begin_inset Formula $C^{1}$
\end_inset

 differentiability in an atlas induces a 
\begin_inset Formula $C^{\infty}$
\end_inset

 atlas as a subset of the original atlas.
 This is why we call a manifold with a differentiable atlas a 
\emph on
smooth manifold
\emph default
.
\end_layout

\begin_layout Enumerate
If we would like to measure distances on 
\begin_inset Formula $\mathbb{M}$
\end_inset

, we introduce a 
\emph on
metric
\emph default
 
\begin_inset Formula $g:\mathbb{M}\times\mathbb{M}\rightarrow[0,\infty)$
\end_inset

 that assigns a distance to each pair of points that remains identical when
 the arguments are swapped, is zero if and only if they are the same point,
 and fulfills the triangle inequality.
 A manifold with a metric is called a 
\emph on
Riemannian manifold
\emph default

\begin_inset Foot
status open

\begin_layout Plain Layout
The concept can be generalized to a 
\emph on
pseudo-Riemannian
\emph default
 manifold for spacetime allowing negative 
\begin_inset Quotes erd
\end_inset

distances
\begin_inset Quotes erd
\end_inset


\end_layout

\end_inset

 
\begin_inset Formula $(\mathbb{M},\mathcal{O},\mathcal{A},g)$
\end_inset

.
\end_layout

\begin_layout Standard
Depending on the required features we can define mathematical objects that
 represent physical quantities at different structural levels.
 Most of the time we require a smooth manifold after step 3.
 If we can avoid the use of the metric structure in step 4 to a bare minimum
 we can gain insight into laws of physics and guidance for discretized numerical
 modelling.
 This will imply keeping vectors and covectors as distinct objects rather
 than converting between the two using the metric tensor.
\end_layout

\begin_layout Subsubsection*
Vectors and covectors
\end_layout

\begin_layout Standard

\series bold
TODO: mention curves
\end_layout

\begin_layout Standard
On a 
\begin_inset Formula $d$
\end_inset

-dimensional smooth manifold we can introduce 
\emph on
vectors
\emph default
 
\begin_inset Formula $\v v\in T_{X}\mathbb{M}$
\end_inset

 of the 
\emph on
tangent space 
\emph default

\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

.
 Intuitively 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 can be imagined as the tangent plane to a curved surface representing the
 manifold.
 The subscript 
\begin_inset Formula $X$
\end_inset

 in 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 indicates that a vector 
\begin_inset Formula $\v v$
\end_inset

 only has a meaning with respect to a specific point 
\begin_inset Formula $X$
\end_inset

, since 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 is generally not the same at different points.
 A further abstraction concerns the basis of the tangent space.
 While we are used to a geometrical definition based on Cartesian coordinates,
 the mathematical definition is more abstract and based on a question what
 a vector can 
\emph on
do
\emph default
.
 This is closely related to the high-school definition of a vector 
\begin_inset Formula $\v v$
\end_inset

, defining it as an arrow pointing from point 
\begin_inset Formula $X_{A}$
\end_inset

 to 
\begin_inset Formula $X_{B}$
\end_inset

.
 In fact we are asking for the difference of a scalar field 
\begin_inset Formula $f$
\end_inset

 if we go along 
\begin_inset Formula $\v v$
\end_inset

 from 
\begin_inset Formula $X_{A}$
\end_inset

 to 
\begin_inset Formula $X_{B}$
\end_inset

.
 Now we are in a situation that allows to do this only in an 
\emph on
infinitesimal neighbourhood
\emph default
 of 
\begin_inset Formula $X$
\end_inset

.
 This provides the 
\emph on
directional derivative along
\emph default
 
\begin_inset Formula $\v v$
\end_inset

 of a scalar field 
\begin_inset Formula $f(X)$
\end_inset

 in a certain chart.
 In classical tensor calculus one would write this expression as
\begin_inset Formula 
\begin{equation}
(\v v\cdot\nabla)f=\left(v^{i}\v e_{i}\cdot\v e^{k}\frac{\partial}{\partial x^{k}}\right)f=\left(v^{i}\delta_{i}^{k}\frac{\partial}{\partial x^{k}}\right)f=\left(v^{k}\frac{\partial}{\partial x^{k}}\right)f=v^{k}\frac{\partial f}{\partial x^{k}}.
\end{equation}

\end_inset

Here we have put brackets to avoid interpretation as a gradient, which will
 later be associated to covectors.
 We realize that we do not require geometrical basis vectors 
\begin_inset Formula 
\begin{equation}
\v e_{k}=\frac{\partial\v r}{\partial x^{k}},
\end{equation}

\end_inset

since they cancel out.
 We can rather introduce an abstract basis 
\begin_inset Formula 
\begin{equation}
\v e_{k}\equiv\frac{\v{\partial}}{\v{\partial}x^{k}}\equiv\v{\partial}_{k}.
\end{equation}

\end_inset

where we have stripped away the reference to the 
\begin_inset Quotes eld
\end_inset

vector
\begin_inset Quotes erd
\end_inset

 
\begin_inset Formula $\v r$
\end_inset

 that has bugged us in tensor calculus, and wrote the 
\begin_inset Formula $\v{\partial}$
\end_inset

 symbols in bold face to identify 
\begin_inset Formula $\v{\partial}/\v{\partial}x^{k}$
\end_inset

 as a vector and distinguish it from a usual partial differential operator.
 Vectors are represented by
\begin_inset Formula 
\begin{equation}
\v v=v^{k}\v{\partial}_{k},
\end{equation}

\end_inset

This means that now vectors can actually 
\emph on
act
\emph default
 on scalar fields with the usual gradient in any chart,
\begin_inset Formula 
\begin{equation}
\v v[f(X)]\equiv v^{k}\frac{\partial f}{\partial x^{k}}.
\end{equation}

\end_inset

Basis vectors 
\begin_inset Formula $\v{\partial}_{k}$
\end_inset

 acting on 
\begin_inset Formula $f$
\end_inset

 yield directional derivatives
\begin_inset Formula 
\begin{equation}
\v{\partial}_{k}[f]=\frac{\partial f}{\partial x^{k}}
\end{equation}

\end_inset

 along curves in direction of 
\begin_inset Formula $x^{k}$
\end_inset

.
\end_layout

\begin_layout Standard
The generalization of the inner product between vectors leads to the concept
 of 
\emph on
covectors 
\emph default
or 
\emph on
one-forms
\emph default
 
\begin_inset Formula $\v w\in T_{X}^{\star}\mathbb{M}$
\end_inset

.
 Here 
\begin_inset Formula $T_{X}^{\star}\mathbb{M}$
\end_inset

 is called the 
\emph on
cotangent space 
\emph default
at 
\begin_inset Formula $X$
\end_inset

 and a covectors form the space of 
\emph on
linear operators
\emph default
 that can be 
\emph on
applied
\emph default
 to a vector 
\begin_inset Formula $\v v$
\end_inset

 to yield a scalar.
 Gradients of scalar fields yield covectors, which can be seen in the classical
 definition of a scalar differential
\begin_inset Formula 
\begin{equation}
\d f=\left(\frac{\partial f}{\partial x^{i}}\v e^{i}\right)\cdot\left(\v e_{k}\d x^{k}\right)=\frac{\partial f}{\partial x^{i}}\delta_{k}^{i}\d x^{k}=\frac{\partial f}{\partial x^{k}}\d x^{k}.
\end{equation}

\end_inset

containing components of the gradient 
\begin_inset Formula $\nabla f$
\end_inset

 as coefficients.
 Now instead of defining 
\begin_inset Formula 
\begin{equation}
\v e^{i}=\nabla x^{i}
\end{equation}

\end_inset

we again 
\begin_inset Quotes eld
\end_inset

strip away
\begin_inset Quotes erd
\end_inset

 the reference to 
\begin_inset Formula $\v r$
\end_inset

 hidden in the gradient and directly define 
\begin_inset Formula 
\begin{equation}
\v e^{i}\equiv\v dx^{i}
\end{equation}

\end_inset

via differentials.
 Covectors are expanded as 
\begin_inset Formula 
\begin{equation}
\v w=w_{i}\v dx^{i}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
A maybe surprising consequence of the separation into vectors and covectors
 is the fact that there exists no inner product 
\emph on
between
\emph default
 the two.
 Instead a covector 
\begin_inset Formula $\v w$
\end_inset

 can 
\emph on
act 
\emph default
on a vector 
\begin_inset Formula $\v v$
\end_inset

 as 
\begin_inset Formula 
\begin{equation}
\v w[\v v]=w_{i}v^{i},
\end{equation}

\end_inset

yielding the same scalar result independently from the chosen coordinate
 chart.
 In particular
\begin_inset Formula 
\begin{equation}
\v dx^{i}\left[\v{\partial}_{k}\right]=\delta_{k}^{i}
\end{equation}

\end_inset

remains valid for basis vectors.
 When skipping brackets and formally writing 
\begin_inset Formula $\v w[\v v]=\v w\cdot\v v$
\end_inset

 as a scalar product one should write the covector 
\begin_inset Formula $\v w$
\end_inset

 first.
 Vectors and covectors thus exist as distinct mathematical objects, even
 though 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 and 
\begin_inset Formula $T_{X}^{\star}\mathbb{M}$
\end_inset

 are isomorphic as finite-dimensional vector spaces of equal dimension 
\begin_inset Formula $d$
\end_inset

.
 A visual representation of basis vectors, covectors and 2-forms is available
 in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:manifold"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename fig/manifold2.pdf
	scale 120

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Local basis 
\begin_inset Formula $\v{\partial}/\v{\partial}\ph$
\end_inset

 for vectors, 
\begin_inset Formula $\v d\ph$
\end_inset

 for covectors, and 
\begin_inset Formula $\v d\theta\wedge\v d\ph$
\end_inset

 for 2-forms.
 While a vector can be interpreted as an arrow originating at a point, covectors
 appear as rates of change on lines connecting two points.
 A 2-form describes a flux through a 2D cell bounded by line elements.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:manifold"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection*
Tensor fields and forms
\end_layout

\begin_layout Standard
Starting from the base manifold 
\begin_inset Formula $\mathbb{M}$
\end_inset

 and "attaching" the tangent space 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 in each points 
\begin_inset Formula $X$
\end_inset

 (actually it just means taking the disjoint union 
\begin_inset Quotes eld
\end_inset


\begin_inset Formula $\bigsqcup$
\end_inset


\begin_inset Quotes erd
\end_inset

 of all tangent spaces) yields the 
\begin_inset Formula $2N$
\end_inset

-dimensional 
\emph on
tangent bundle
\series bold
 
\begin_inset Formula 
\begin{equation}
T\mathbb{M}\equiv\bigsqcup_{X\in\mathbb{M}}T_{X}\mathbb{M}
\end{equation}

\end_inset


\series default
\emph default
whose elements are tuples 
\begin_inset Formula $(X,\v v)$
\end_inset

 of possible combinations of a position 
\begin_inset Formula $X\in\mathbb{M}$
\end_inset

 and a vector 
\begin_inset Formula $\v v\in T_{X}\mathbb{M}$
\end_inset

 tangent to 
\begin_inset Formula $\mathbb{M}$
\end_inset

 at 
\begin_inset Formula $X$
\end_inset

.
 If 
\begin_inset Formula $\v v$
\end_inset

 has a specific functional dependency on 
\begin_inset Formula $X$
\end_inset

 in these tuples our first try would be to just define 
\begin_inset Formula $\v v(X)$
\end_inset

 as a vector-valued field on 
\begin_inset Formula $\mathbb{M}$
\end_inset

.
 Since the vector space 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset

 where 
\begin_inset Formula $\v v(X)$
\end_inset

 points to depends on the argument 
\begin_inset Formula $X$
\end_inset

, we require a more technical construction.
 We define a 
\emph on
vector field 
\emph default

\begin_inset Formula 
\begin{align}
\v V: & \mathbb{M}\rightarrow T\mathbb{M},\nonumber \\
 & X\mapsto(X,\v v(X))
\end{align}

\end_inset

that keeps the information of 
\begin_inset Formula $X$
\end_inset

 in the target 
\begin_inset Formula $(X,\v v(X))\in T\mathbb{M}$
\end_inset

.
 Such a map is called a 
\emph on
section
\emph default
 of 
\series bold
\emph on

\begin_inset Formula $T\mathbb{M}$
\end_inset


\series default
\emph default
 and is the analogue to the graph 
\begin_inset Formula $(x,y(x))$
\end_inset

 of a function 
\begin_inset Formula $y(x)$
\end_inset

.
 Usually we ignore the technical detail of writing 
\begin_inset Formula $(X,\v v(X))$
\end_inset

, since 
\begin_inset Formula $X$
\end_inset

 appears as an argument anyway.
\end_layout

\begin_layout Standard
Similarly, attaching cotangent spaces 
\begin_inset Formula $T_{X}^{\star}\mathbb{M}$
\end_inset

 in each point on 
\begin_inset Formula $\mathbb{M}$
\end_inset

 results in the 
\emph on
cotangent bundle 
\emph default

\begin_inset Formula $T^{\star}\mathbb{M}$
\end_inset

 containing pairs 
\begin_inset Formula $(X,\v w)$
\end_inset

.
 
\emph on
Covector fields
\emph default
 
\begin_inset Formula $\v W:\mathbb{M}\rightarrow T^{\star}\mathbb{M}$
\end_inset

 are specified via sections 
\begin_inset Formula $(X,\v w(X))$
\end_inset

 of the cotangent bundle 
\begin_inset Formula $T^{\star}\mathbb{M}$
\end_inset

.
 Similar to vectors and covectors, a covector field acts on a vector field
 via
\begin_inset Formula 
\begin{equation}
\v w(X)[\v v(X)]=w_{i}(X)v^{i}(X),
\end{equation}

\end_inset

where the point 
\begin_inset Formula $X$
\end_inset

 appears as an additional argument and can be specified in any suitable
 set of coordinates 
\begin_inset Formula $\xset$
\end_inset

 in a chart via the chart map 
\begin_inset Formula $X(\xset)$
\end_inset

.
 We are only interested in sufficiently smooth vector fields, where components
 
\begin_inset Formula $v^{i}(\xset)\equiv v^{i}(X(\xset))$
\end_inset

 and 
\begin_inset Formula $w_{i}(\xset)\equiv w_{i}(X(\xset))$
\end_inset

 are sufficiently smooth maps from 
\begin_inset Formula $\mathbb{R}^{d}$
\end_inset

 to 
\begin_inset Formula $\mathbb{R}^{d}$
\end_inset

.
 Note that up to now we didn't require a metric tensor.
 
\series bold
TODO: mention tensor fields and forms
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Maps and their properties
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Map
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
from
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
to
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
space
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
remark
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Scalar field 
\begin_inset Formula $\Phi$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{R}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $C^{\infty}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Vector 
\begin_inset Formula $\v v$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $C^{\infty}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{R}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Covector 
\begin_inset Formula $\v w$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{R}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T_{X}^{\star}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
linear
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Vector field 
\begin_inset Formula $(X,\v v(X))$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\Gamma(T\mathbb{M})$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\v v(X)$
\end_inset

 is a vector in 
\begin_inset Formula $T_{X}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Covector field 
\begin_inset Formula $(X,\v w(X))$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T^{\star}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\Gamma(T^{\star}\mathbb{M})$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\v w(X)$
\end_inset

 is a covector in 
\begin_inset Formula $T_{X}^{\star}\mathbb{M}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection*
The metric tensor
\end_layout

\begin_layout Standard
On manifolds with a metric 
\begin_inset Formula $g$
\end_inset

, the 
\emph on
metric tensor
\emph default
 
\begin_inset Formula $\mathbf{g}$
\end_inset

 provides a natural isomorphism to between vectors and covectors.
 More precisely, 
\begin_inset Formula $\t g$
\end_inset

 is a symmetric 
\begin_inset Formula $(0,2)$
\end_inset

-tensor field (that can depend on position 
\begin_inset Formula $\v q$
\end_inset

) on 
\begin_inset Formula $Q$
\end_inset

 with components 
\begin_inset Formula $g_{ik}=g_{ik}(\v q)$
\end_inset

 in a certain coordinate chart.
 The action of 
\begin_inset Formula $\mathbf{g}$
\end_inset

 on two vectors is the commonly known 
\emph on
inner product 
\begin_inset Formula 
\begin{equation}
\v u\cdot\v v\equiv\mathbf{g}(\v u,\v v)=\mathbf{g}(\v v,\v u)=u^{i}g_{ik}v^{k}=v^{i}g_{ik}u^{k},\label{eq:guv}
\end{equation}

\end_inset


\emph default
which is a 
\emph on
symmetric
\emph default
 
\emph on
bilinear
\emph default
 form, i.e.
 linear in both arguments 
\begin_inset Formula $\v u$
\end_inset

 and 
\begin_inset Formula $\v v$
\end_inset

 which can be swapped.
 Similarly the inner product in cotangent space
\begin_inset Formula 
\begin{equation}
\v p\cdot\v w\equiv\bar{\mathbf{g}}(\v p,\v w)=\bar{\mathbf{g}}(\v w,\v p)=p_{j}\bar{g}^{jl}w_{l}=w_{j}\bar{g}^{jl}p_{l}
\end{equation}

\end_inset

is provided via the 
\emph on
reciprocal metric
\emph default
 
\begin_inset Formula $\bar{\mathbf{g}}$
\end_inset

 being a symmetric 
\begin_inset Formula $(2,0)$
\end_inset

-tensor field on 
\begin_inset Formula $Q$
\end_inset

 defined by
\begin_inset Formula 
\begin{equation}
\bar{g}^{ij}g_{jk}=\delta_{k}^{i}\equiv\begin{cases}
0 & \text{for }i\neq k,\\
1 & \text{for\,}i=k
\end{cases}
\end{equation}

\end_inset

This means that the component matrix 
\begin_inset Formula $(\bar{g}^{jl})=(g_{ik})^{-1}$
\end_inset

 is the inverse matrix of the metric tensor component matrix in any set
 of coordinates.
 It is important to emphasize that 
\begin_inset Formula $\mathbf{g}$
\end_inset

 and 
\begin_inset Formula $\bar{\mathbf{g}}$
\end_inset

 are not the same tensor field, even though the common notation 
\begin_inset Formula $g^{jl}$
\end_inset

 without a bar would suggest this.
 Since 
\begin_inset Formula $u_{k}\equiv u^{i}g_{ik}$
\end_inset

 are components of a covector dual to 
\begin_inset Formula $\v u$
\end_inset

 and 
\begin_inset Formula $w^{l}\equiv w_{j}\bar{g}^{jl}$
\end_inset

 the ones of a vector dual to 
\begin_inset Formula $\v w$
\end_inset

, one can 
\begin_inset Quotes eld
\end_inset

convert
\begin_inset Quotes erd
\end_inset

 between vectors (contravariant representation) and covectors (covariant
 representation) by raising and lowering indexes via the (inverse) metric.
 This allows the reduction to only vectors as fundamental objects, as used
 in traditional tensor calculus.
 If quantities are represented in their 
\begin_inset Quotes eld
\end_inset

natural
\begin_inset Quotes erd
\end_inset

 way in the traditional formalism, i.e.
 contravariant components for tangent vectors, the two and covariant components
 for gradient 
\begin_inset Quotes eld
\end_inset

vectors
\begin_inset Quotes erd
\end_inset

, formulas become equivalent to the differential geometrical variant, since
 the metric tensor disappears.
 As soon as divergence and curl operators enter, a description using either
 differential forms or tensor densities is required to be able to work in
 a metric-independent way.
\end_layout

\begin_layout Section
Mechanics
\end_layout

\begin_layout Standard
In classical mechanics the phase-space 
\begin_inset Formula $\mathbb{M}$
\end_inset

 is a 
\begin_inset Formula $2N$
\end_inset

-dimensional symplectic manifold constructed from combination of positions
 
\begin_inset Formula $Q$
\end_inset

 being elements of an 
\begin_inset Formula $N$
\end_inset

-dimensional configuration manifold 
\begin_inset Formula $\mathbb{Q}$
\end_inset

 with their respective momenta as covectors 
\begin_inset Formula $\v p$
\end_inset

 in the 
\begin_inset Formula $N$
\end_inset

-dimensional cotangent space 
\begin_inset Formula $T_{Q}^{\star}\mathbb{Q}$
\end_inset

 at position 
\begin_inset Formula $Q$
\end_inset

.
 As defined above 
\begin_inset Formula $T_{Q}^{\star}\mathbb{Q}$
\end_inset

 is the cotangent space dual to the tangent space 
\begin_inset Formula $T_{Q}\mathbb{Q}$
\end_inset

, where the latter contains velocity vectors 
\begin_inset Formula $\dot{\v q}$
\end_inset

.
 Here the notation 
\begin_inset Formula $\dot{\v q}$
\end_inset

 for velocity is conventional 
\emph on
a priori
\emph default
 and must not be identified with time derivatives 
\begin_inset Formula $\d Q(t)/\d t$
\end_inset

 appearing only for a specific orbit curve 
\begin_inset Formula $Q(t)$
\end_inset

.
 The 
\emph on
Lagrangian
\emph default
 formalism in 
\begin_inset Formula $(Q,\dot{\v q})$
\end_inset

 is constructed on the tangent bundle 
\begin_inset Formula $T\mathbb{Q}$
\end_inset

 of 
\begin_inset Formula $\mathbb{Q}$
\end_inset

, while the 
\emph on
Hamiltonian 
\emph default
formalism is stated on the cotangent bundle 
\begin_inset Formula $T^{\star}\mathbb{Q}$
\end_inset

.
 In terms of notation the outlined construction leads to upper i.e.
 contravariant index notation for generalized coordinates 
\begin_inset Formula $q^{i}$
\end_inset

 representing points 
\begin_inset Formula $Q$
\end_inset

 in a certain chart, and for velocity components 
\begin_inset Formula $\dot{q}^{i}$
\end_inset

 in the according local chart basis of 
\begin_inset Formula $T_{Q}\mathbb{Q}$
\end_inset

 at 
\begin_inset Formula $Q$
\end_inset

.
 Lower indexes are used for 
\begin_inset Formula $p_{j}$
\end_inset

 being covariant components of 
\begin_inset Formula $\v p$
\end_inset

 with respect to the local chart basis of 
\begin_inset Formula $T_{Q}^{\star}\mathbb{Q}$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
General phase-space
\end_layout

\begin_layout Standard
Phase-space 
\begin_inset Formula $\mathbb{M}=T^{\star}\mathbb{Q}$
\end_inset

 still allows for a clear distinction of 
\begin_inset Formula $Q$
\end_inset

 and 
\begin_inset Formula $\v p$
\end_inset

 as different mathematical objects and underlies the canonical 
\emph on
Hamiltonian
\emph default
 formalism in 
\begin_inset Formula $(Q,\v p)$
\end_inset

.
 For a more general non-canonical treatment it is however useful to introduce
 points 
\begin_inset Formula $Z\in\mathbb{M}$
\end_inset

 independently from the underlying canonical foundation.
 This makes the Hamiltonian formalism applicable to phase-spaces 
\begin_inset Formula $\mathbb{M}$
\end_inset

 that haven't been constructed as a cotangent bundle and in this sense more
 general than the Lagrangian formalism.
 The Hamiltonian formalism in possibly non-canonical coordinates is a special
 case of the even more general treatment via the 
\emph on
phase-space Lagrangian
\emph default
 on the 
\begin_inset Formula $4N$
\end_inset

-dimensional tangent bundle 
\begin_inset Formula $T\mathbb{M}$
\end_inset

 of 
\begin_inset Formula $\mathbb{M}$
\end_inset

, being the tangent bundle 
\begin_inset Formula $T(T^{\star}\mathbb{Q})$
\end_inset

 of the cotangent bundle of 
\begin_inset Formula $\mathbb{Q}$
\end_inset

 in classical mechanics.
 Elements 
\begin_inset Formula $(Z,\dot{\v z})$
\end_inset

 of 
\begin_inset Formula $T\mathbb{M}$
\end_inset

 are treated analogously to 
\begin_inset Formula $(Q,\dot{\v q})$
\end_inset

 in the usual Lagrangian formalism.
 Thus, coordinates 
\begin_inset Formula $z^{\alpha}$
\end_inset

 are written with upper index 
\begin_inset Formula $\alpha=1\dots2N$
\end_inset

, ignoring the previous separation in 
\begin_inset Formula $q^{i}$
\end_inset

 and 
\begin_inset Formula $p_{j}$
\end_inset

 being 
\emph on
canonical
\emph default
 coordinates for 
\begin_inset Formula $\mathbb{M}$
\end_inset

, but also individual coordinates for 
\begin_inset Formula $\mathbb{Q}$
\end_inset

 and 
\begin_inset Formula $T^{\star}\mathbb{Q}$
\end_inset

, respectively.
 Euler-Lagrange equations of the phase-space Lagrangian 
\begin_inset Formula $L_{\mathrm{ph}}(Z,\dot{\v z},t)$
\end_inset

 correspond to Hamiltonian equations of motion for Hamiltonian 
\begin_inset Formula $H(Z,t)$
\end_inset

 in possibly non-canonical coordinates as long as the canonical representation
 of 
\begin_inset Formula $L_{\mathrm{ph}}$
\end_inset

 is of the form 
\begin_inset Formula 
\begin{equation}
L_{\mathrm{ph}}(Z,\dot{\v z},t)=p_{k}\dot{q}^{k}-H(Z,t).
\end{equation}

\end_inset

This means that no components 
\begin_inset Formula $\dot{p}_{k}$
\end_inset

 of momentum components of tangent vectors 
\begin_inset Formula $\dot{\v z}$
\end_inset

 must appear in 
\begin_inset Formula $L_{\mathrm{ph}}$
\end_inset

.
 This feature is known as the 
\emph on
degeneracy
\emph default
 of the phase-space Lagrangian and prevents the construction of a hypothetical
 higher-level "phase-space-Hamiltonian" formalism on 
\begin_inset Formula $T^{\star}\mathbb{M}$
\end_inset

 via a Legendre transform.
 A more elegant representation of the phase-space Lagrangian is achieved
 by 
\begin_inset Quotes eld
\end_inset

stripping away
\begin_inset Quotes erd
\end_inset

 
\begin_inset Formula $\d t$
\end_inset

 via multiplication and writing it as the Lagrangian one-form
\begin_inset Formula 
\begin{equation}
\v{\theta}_{L}(Z,t)\equiv p_{k}\v dq^{k}-H(Z,t)\v dt.
\end{equation}

\end_inset

Since this one-form is now defined on the manifold 
\begin_inset Formula $\mathbb{R}\times\mathbb{M}$
\end_inset

 including 
\begin_inset Formula $t\in\mathbb{R}$
\end_inset

 we have added a bold differential 
\begin_inset Formula $\v dt$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
Symplectic structure
\end_layout

\begin_layout Standard

\series bold
TODO: check sign errors
\end_layout

\begin_layout Standard
The 2-form 
\begin_inset Formula $\v{\omega}$
\end_inset

 is an 
\emph on
antisymmetric
\emph default
 
\emph on

\begin_inset Formula $(0,2)$
\end_inset


\emph default
-tensor field and can be seen as an antisymmetric analogue to the metric
 tensor 
\begin_inset Formula $\mathbf{g}$
\end_inset

.
 While also a bilinear form, instead of the symmetric relation in Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:guv"

\end_inset

 it switches sign when argument vectors 
\begin_inset Formula $\v y,\v z$
\end_inset

 are swapped, i.e.
\begin_inset Formula 
\begin{equation}
\v{\omega}(\v y,\v z)=-\v{\omega}(\v z,\v y)=y^{\alpha}\omega_{\alpha\beta}z^{\beta}=-z^{\alpha}\omega_{\alpha\beta}y^{\beta}.
\end{equation}

\end_inset

In the context of Hamiltonian systems components 
\begin_inset Formula $\omega_{jk}$
\end_inset

 are represented by the inverse matrix of the Poisson matrix 
\begin_inset Formula $\Pi^{ij}$
\end_inset

 representing the Poisson tensor 
\begin_inset Formula $\v{\Pi}$
\end_inset

 that appears in equations of motion
\begin_inset Formula 
\begin{equation}
\dot{z}^{\alpha}=\Pi^{\alpha\beta}\frac{\partial H}{\partial z^{\beta}}.\label{eq:zadot}
\end{equation}

\end_inset

One could say that 
\begin_inset Formula $\v{\Pi}$
\end_inset

 takes over the role of the inverse metric tensor 
\begin_inset Formula $\bar{\mathbf{g}}$
\end_inset

 in the symplectic analogy.
 Before introducing a particular set of coordinates we emphasize that it
 is important to distinguish invariant objects such as 
\begin_inset Formula $\v{\omega},\v{\Pi}$
\end_inset

 from their coordinate representation 
\begin_inset Formula $\omega_{ij},\Pi^{ij}$
\end_inset

.
 The fact that it is possible to 
\emph on
locally
\emph default
 find canonical coordinates in any manifold 
\begin_inset Formula $\mathbb{M}$
\end_inset

 with a symplectic structure 
\begin_inset Formula $\v{\omega}$
\end_inset

 is known as 
\emph on
Darboux's theorem
\emph default
.
 This is the analogy to the possibility to find a locally flat representation
 of a curved manifold with a metric structure 
\begin_inset Formula $\t g$
\end_inset

, which is important in general relativity.
\end_layout

\begin_layout Standard
In any phase-space coordinate basis
\begin_inset Formula 
\begin{equation}
\v e_{\alpha}\equiv\frac{\v{\partial}}{\v{\partial}z^{\alpha}}\equiv\v{\partial}_{\alpha}
\end{equation}

\end_inset

we can expand the Poisson tensor 
\begin_inset Formula $\v{\Pi}$
\end_inset

 as
\begin_inset Formula 
\begin{equation}
\v{\Pi}=\Pi^{\alpha\beta}\v{\partial}_{\alpha}\otimes\v{\partial}_{\beta},
\end{equation}

\end_inset

using the Poisson matrix 
\begin_inset Formula $(\Pi^{\alpha\beta})$
\end_inset

.
 Since the second unit vector 
\begin_inset Formula $\v{\partial}_{\beta}$
\end_inset

 can on a scalar field, in particular on 
\begin_inset Formula $H$
\end_inset

, one can denote equations of motion 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:zadot"

\end_inset

 in a coordinate free way via
\begin_inset Formula 
\begin{equation}
\dot{\v z}[f]=\v{\Pi}[f,H].\label{eq:zadot-1}
\end{equation}

\end_inset

In canonical coordinates 
\begin_inset Formula $z_{\mathrm{can}}^{\alpha}\equiv q^{i},p_{j}$
\end_inset

 the Poisson matrix becomes the totally antisymmetric matrix
\begin_inset Formula 
\begin{equation}
\Pi^{\alpha\beta}=J^{\alpha\beta}=\left(\begin{array}{cc}
0 & I\\
-I & 0
\end{array}\right)\quad\text{for}\,z_{\mathrm{can}}^{\alpha},
\end{equation}

\end_inset

where 
\begin_inset Formula $I$
\end_inset

 is the identity matrix of dimension 
\begin_inset Formula $N$
\end_inset

.
 Thus we can write the identity
\begin_inset Formula 
\begin{align}
\v{\Pi} & =\delta_{k}^{i}\frac{\v{\partial}}{\v{\partial}q^{i}}\otimes\frac{\v{\partial}}{\v{\partial}p_{k}}-\delta_{k}^{i}\frac{\v{\partial}}{\v{\partial}p_{k}}\otimes\frac{\v{\partial}}{\v{\partial}q^{i}}\nonumber \\
 & =\frac{\v{\partial}}{\v{\partial}q^{i}}\otimes\frac{\v{\partial}}{\v{\partial}p_{i}}-\frac{\v{\partial}}{\v{\partial}p_{i}}\otimes\frac{\v{\partial}}{\v{\partial}q^{i}}.
\end{align}

\end_inset

in any set of canonical coordinates, where sums over 
\begin_inset Formula $i,k$
\end_inset

 run from 
\begin_inset Formula $1$
\end_inset

 to 
\begin_inset Formula $N$
\end_inset

.
 This finally recovers canonical equations of motion
\begin_inset Formula 
\begin{equation}
\dot{q}^{i}=\frac{\partial H}{\partial p_{i}},\quad\dot{p}_{i}=\frac{\partial H}{\partial q^{i}}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Components of the symplectic form
\begin_inset Formula 
\[
\v{\omega}=\omega_{\alpha\beta}\,\v dz^{\alpha}\wedge\v dz^{\beta}
\]

\end_inset

with respect to the dual basis
\begin_inset Formula 
\begin{equation}
\v e^{\alpha}\equiv\v dz^{\alpha}
\end{equation}

\end_inset

are defined such that
\begin_inset Formula 
\begin{equation}
\omega_{\alpha\gamma}\Pi^{\gamma\beta}=\delta_{\alpha}^{\beta}
\end{equation}

\end_inset

in any system of coordinates in phase-space.
 Components 
\begin_inset Formula $\omega_{\alpha\beta}$
\end_inset

 are sometimes called the Lagrange matrix
\emph on
 
\emph default
which is the inverse of the Poisson matrix 
\begin_inset Formula $\Pi^{\alpha\beta}$
\end_inset

.
 Showing the invariance of the symplectic form 
\begin_inset Formula $\v{\omega}$
\end_inset

 under the numerical flow map generated by an integrator is thus equivalent
 to showing the invariance of 
\begin_inset Formula $\v{\Pi}$
\end_inset

.
 Multiplying equations of motion
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:zadot"

\end_inset

 from the left via 
\begin_inset Formula $\omega_{\alpha\beta}$
\end_inset

 and renaming indexes yields the equivalent system
\begin_inset Formula 
\begin{equation}
\omega_{\alpha\beta}\dot{z}^{\beta}=\frac{\partial H}{\partial z^{\alpha}}.\label{eq:zadot-2}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
In particular in canonical coordinates we have
\begin_inset Formula 
\begin{align}
\omega_{\alpha\beta} & =(\Pi^{\alpha\beta})^{-1}\nonumber \\
 & =(J^{\alpha\beta})^{-1}=\left(\begin{array}{cc}
0 & -I\\
I & 0
\end{array}\right)\quad\text{for}\,z_{\mathrm{can}}^{i}.\label{eq:cani}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
We introduce the antisymmetric wedge product as a basis for 2-forms, i.e.
 antisymmetric rank-2 tensor fields in covariant representation.
 The space of 
\begin_inset Formula $2$
\end_inset

-forms can be seen as a subspace of the space of all rank-2 tensor fields
 spanned by
\begin_inset Formula 
\begin{equation}
\v dz^{i}\otimes\v dz^{j},\text{ i.e.}\,(\v dq^{i}\otimes\v dq^{j},\v dq^{i}\otimes\v dp_{j},\v dp_{i}\otimes\v dq^{j},\v dp_{i}\otimes\v dp_{j}),
\end{equation}

\end_inset

where the wedge notation allows only for antisymmetric tensor fields via
\begin_inset Formula 
\begin{equation}
\v dq^{i}\wedge\v dp_{j}\equiv\v dq^{i}\otimes\v dp_{j}-\v dp_{i}\otimes\v dq^{j}.
\end{equation}

\end_inset

 This means that the wedge product fulfills the antisymmetry relations
\begin_inset Formula 
\begin{align}
\v dq^{i}\wedge\v dp_{j}=-\v dq^{j}\wedge\v dp_{i} & .
\end{align}

\end_inset

Finally, according to Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:cani"

\end_inset

 we can express the symplectic 2-form 
\begin_inset Formula $\v{\omega}$
\end_inset

 via the sum over canonical wedge tuples,
\begin_inset Formula 
\begin{equation}
\v{\omega}=-\delta_{i}^{j}\v dp_{i}\otimes\v dq^{j}+\delta_{j}^{i}\v dq^{i}\otimes\v dp_{j}=\delta_{j}^{i}\v dq^{i}\wedge\v dp_{j}=\v dq^{i}\wedge\v dp_{i}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\series bold
TODO:
\end_layout

\begin_layout Standard
Variants of the special 2-form 
\begin_inset Formula $\v J$
\end_inset

 correspond to vectors in a similar way as diagonal matrices being generalizatio
ns of the identity matrix 
\begin_inset Formula $\v I$
\end_inset

 correspond to vectors containing only diagonal elements.
 Make some connection to the two representations of 
\begin_inset Formula $\v B$
\end_inset

 as a vector density and a 2-form.
 The idea comes from the incomplete eigendecomposition where 
\begin_inset Formula $\v{\lambda}_{m}\hat{V}_{m}=\hat{V}_{m}\hat{\Lambda}_{mm}=\hat{V}_{m}\hat{I}_{m}\v{\lambda}_{m}$
\end_inset


\end_layout

\begin_layout Section
Forms and densities
\end_layout

\begin_layout Standard
or: "why vector density fields are the same as 2-forms and what this means
 for the magnetic field".
 The reason for this section is the fact that physicists take divergence
 and curl of vector fields, while mathematicians use them on differential
 forms.
 Tensor densities provide a link between the two, yielding identical coefficient
s of a 2-form and a contravariant vector density representation, both used
 to describe the magnetic field 
\begin_inset Formula $\v B$
\end_inset

.
 Right now this description requires already some understanding of forms,
 wedge product, and the Hodge star operator.
 
\series bold
TODO: explain more.
\end_layout

\begin_layout Standard
More information on this issue can be found in 
\begin_inset Quotes eld
\end_inset

CONTRAVARIANCE, COVARIANCE, DENSITIES, AND ALL THAT: AN INFORMAL DISCUSSION
 ON TENSOR CALCULUS
\begin_inset Quotes erd
\end_inset

 by Chris Tiee, p.19 - densitization.
 In this sense, also the Levi-Civita symbol appears as a tensor density.
\end_layout

\begin_layout Standard
In classical tensor analysis (see e.g.
 book "Flux coordinates and magnetic field structure" of Callen/d'haeseleer)
 the divergence operator acts on covariant components of a vector field
 and yields a scalar field 
\begin_inset Formula $D$
\end_inset

,
\begin_inset Formula 
\begin{equation}
F=\nabla\cdot\v B\equiv\frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{k}}(\sqrt{g}B^{k}).
\end{equation}

\end_inset

This operator becomes metric-independent if we define a representation of
 
\begin_inset Formula $D$
\end_inset

 via a scalar density 
\begin_inset Formula $\mathcal{D}$
\end_inset

 and a representation of 
\begin_inset Formula $\v B$
\end_inset

 via contravariant density component 
\begin_inset Formula $\mathcal{B}^{k}$
\end_inset

, defined via
\begin_inset Formula 
\begin{equation}
\mathcal{F}\equiv\sqrt{g}F,\quad\mathcal{B}^{k}\equiv\sqrt{g}B^{k}.
\end{equation}

\end_inset

Then we can write the divergence as
\begin_inset Formula 
\begin{equation}
\mathcal{F}=\mathrm{div}\,\v B\equiv\sqrt{g}(\nabla\cdot\v B)=\frac{\partial}{\partial x^{k}}\mathcal{B}^{k}.
\end{equation}

\end_inset

In 3D space, the curl operator acts on covariant components of a vector
 field 
\begin_inset Formula $\v A$
\end_inset


\begin_inset Formula 
\begin{equation}
B^{i}=(\nabla\times\v A)^{i}\equiv\frac{\varepsilon^{ijk}}{\sqrt{g}}\frac{\partial}{\partial x^{j}}A_{k}.
\end{equation}

\end_inset

Using contravariant density components we can write instead a metric-free
 variant
\begin_inset Formula 
\begin{equation}
\mathcal{B}^{i}=(\mathrm{curl}\,\v A)^{i}\equiv\sqrt{g}(\nabla\times\v A)^{i}=\varepsilon^{ijk}\frac{\partial}{\partial x^{j}}A_{k}.
\end{equation}

\end_inset

The divergence of a curl vanishes, since
\begin_inset Formula 
\begin{align}
\mathrm{div}\,\mathrm{curl}\,\v A & =\frac{\partial}{\partial x^{i}}(\varepsilon^{ijk}\frac{\partial}{\partial x^{j}}A_{k})=\varepsilon^{ijk}\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}A_{k}\nonumber \\
 & =-\varepsilon^{jik}\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}A_{k}=-\varepsilon^{ijk}\frac{\partial^{2}}{\partial x^{j}\partial x^{i}}A_{k}\nonumber \\
 & =-\varepsilon^{ijk}\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}A_{k}.\label{eq:zerodiv}
\end{align}

\end_inset

where we have first used the antisymmetric property of 
\begin_inset Formula $\varepsilon^{ijk}$
\end_inset

, then swapped names of indices 
\begin_inset Formula $i$
\end_inset

 and 
\begin_inset Formula $j$
\end_inset

, and finally used the symmetry of second derivatives to swap their order.
 The only way that relation
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:zerodiv"

\end_inset

 with both 
\begin_inset Formula $+$
\end_inset

 and 
\begin_inset Formula $-$
\end_inset

 in front of the same term can be fulfilled, is if it is identically zero.
\end_layout

\begin_layout Standard
Now we are going to show the equivalence of contravariant density representation
 to the representation of 
\begin_inset Formula $\v B$
\end_inset

 as a 2-form.
 We have
\begin_inset Formula 
\begin{equation}
\v{\omega}_{B}=\frac{1}{2}B_{jk}\v dx^{j}\wedge\v dx^{k}=B_{12}\v dx^{1}\wedge\v dx^{2}+B_{23}\v dx^{2}\wedge\v dx^{3}+B_{31}\v dx^{3}\wedge\v dx^{1}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Taking a divergence of this form means to take an exterior derivative
\begin_inset Formula 
\begin{align}
\v{\omega}_{D}=\v d\v{\omega}_{B} & =\frac{1}{2}\v d(B_{jk}\v dx^{j}\wedge\v dx^{k})=\frac{1}{2}\v dB_{jk}\wedge\v dx^{j}\wedge\v dx^{k}\nonumber \\
 & =\frac{1}{2}\frac{\partial B_{jk}}{\partial x^{i}}\v dx^{i}\wedge\v dx^{j}\wedge\v dx^{k}=\frac{1}{2}\varepsilon^{ijk}\frac{\partial B_{jk}}{\partial x^{i}}\v dx^{i}\wedge\v dx^{j}\wedge\v dx^{k}\nonumber \\
 & =\left(\frac{\partial B_{12}}{\partial x^{1}}+\frac{\partial B_{23}}{\partial x^{2}}+\frac{\partial B_{31}}{\partial x^{3}}\right)\v dx^{1}\wedge\v dx^{2}\wedge\v dx^{3}=\omega_{D}\v dx^{1}\wedge\v dx^{2}\wedge\v dx^{3}.
\end{align}

\end_inset

We see that coincides with the divergence defined above via density components,
 if we identify
\begin_inset Formula 
\begin{equation}
\mathcal{D}\equiv\omega_{D},\quad\mathcal{B}^{1}\equiv B_{23},\quad\mathcal{B}^{2}\equiv B_{31},\quad\mathcal{B}^{3}\equiv B_{12}.\label{eq:div}
\end{equation}

\end_inset

This is also consistent the form 
\begin_inset Formula $\v{\omega}_{B}$
\end_inset

 follows from the curl of an 1-form 
\begin_inset Formula $\v A$
\end_inset

 which means
\begin_inset Formula 
\begin{align}
\v{\omega}_{B} & =\v d\v A=\v d(A_{k}\v dx^{k})=\v dA_{k}\wedge\v dx^{k}=\frac{\partial A_{k}}{\partial x^{i}}\v dx^{i}\wedge\v dx^{k}\nonumber \\
 & =\left(\frac{\partial A_{2}}{\partial x^{1}}-\frac{\partial A_{1}}{\partial x^{2}}\right)\v dx^{1}\wedge\v dx^{2}+\left(\frac{\partial A_{3}}{\partial x^{2}}-\frac{\partial A_{2}}{\partial x^{3}}\right)\v dx^{2}\wedge\v dx^{3}+\left(\frac{\partial A_{1}}{\partial x^{3}}-\frac{\partial A_{3}}{\partial x^{1}}\right)\v dx^{3}\wedge\v dx^{1}\nonumber \\
 & =\frac{\partial}{\partial x^{j}}A_{k}\,\v dx^{j}\wedge\v dx^{k}.
\end{align}

\end_inset

We thus can again identify
\begin_inset Formula 
\begin{equation}
\varepsilon^{ijk}B_{jk}=\varepsilon^{ijk}\frac{\partial}{\partial x^{j}}A_{k}=\mathcal{B}^{i}
\end{equation}

\end_inset

with density components of the vector field 
\begin_inset Formula $\v B$
\end_inset

 defined earlier, and consistent with
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:div"

\end_inset

 meaning in particular
\begin_inset Formula 
\begin{align}
B_{12} & =\frac{\partial A_{2}}{\partial x^{1}}-\frac{\partial A_{1}}{\partial x^{2}}=\mathcal{B}^{3},\\
B_{23} & =\frac{\partial A_{3}}{\partial x^{2}}-\frac{\partial A_{2}}{\partial x^{3}}=\mathcal{B}^{1},\\
B_{31} & =\frac{\partial A_{1}}{\partial x^{3}}-\frac{\partial A_{3}}{\partial x^{1}}=\mathcal{B}^{2}.
\end{align}

\end_inset


\end_layout

\begin_layout Standard
We can do the same using the Hodge-star operator, see 
\begin_inset CommandInset href
LatexCommand href
target "https://unapologetic.wordpress.com/2011/10/13/the-divergence-operator/"

\end_inset

, 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

https://en.wikipedia.org/wiki/Hodge_star_operator
\end_layout

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
target "https://lamington.wordpress.com/2014/05/26/div-grad-curl-and-all-this/"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
target "http://www.thp.uni-koeln.de/gravitation/courses/rcii14/fwh.pdf"

\end_inset

.
 Claim: the usual divergence is
\begin_inset Formula 
\begin{equation}
\nabla\cdot\v B=\star(\v d(\star(\v B^{\flat})).\label{eq:hodgestar}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Take the Hodge-star operator for a 
\begin_inset Formula $p$
\end_inset

-form, 
\begin_inset Formula 
\begin{equation}
\star\v{\omega}=\frac{\sqrt{g}}{(n-p)!p!}\varepsilon_{\alpha_{1}\dots\alpha_{p}\beta_{1}\dots\beta_{n-p}}g^{\alpha_{1}\gamma_{1}}\dots g^{\alpha_{p}\gamma_{p}}\omega_{\gamma_{1}\dots\gamma_{p}}\v dx^{\beta_{1}}\wedge\dots\wedge\v dx^{\beta_{n-p}}.
\end{equation}

\end_inset

Then we start with
\begin_inset Formula 
\begin{equation}
\v B=B^{i}\frac{\partial}{\partial x^{i}}
\end{equation}

\end_inset

and represent as a 1-form
\begin_inset Formula 
\begin{equation}
\v B^{\flat}=B_{k}\v dx^{k}=B^{i}g_{ik}\v dx^{k}.
\end{equation}

\end_inset

Then we apply the hodge star
\begin_inset Formula 
\begin{align}
\star\v B^{\flat} & =\frac{\sqrt{g}}{2}\varepsilon_{lmn}g^{lk}B^{i}g_{ik}\v dx^{m}\wedge\v dx^{n}\nonumber \\
 & =\frac{\sqrt{g}}{2}\varepsilon_{lmn}\delta_{i}^{l}B^{i}\v dx^{m}\wedge\v dx^{n}\nonumber \\
 & =\frac{1}{2}\varepsilon_{lmn}\mathcal{B}^{l}\,\v dx^{m}\wedge\v dx^{n}\nonumber \\
 & =\mathcal{B}^{1}\,\v dx^{2}\wedge\v dx^{3}+\mathcal{B}^{2}\,\v dx^{3}\wedge\v dx^{1}+\mathcal{B}^{3}\,\v dx^{1}\wedge\v dx^{2}=\v{\omega}_{B}.
\end{align}

\end_inset

Here the metric and the inverse metric could be cancelled to a 
\begin_inset Formula $\delta$
\end_inset

 via contraction.
 Then we apply an exterior derivative, and finally taking another hodge
 
\begin_inset Formula $\star$
\end_inset

 yields a scalar field instead of a density in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:hodgestar"

\end_inset

.
 The same works for the curl (see 
\begin_inset CommandInset href
LatexCommand href
target "https://en.wikipedia.org/wiki/Curl_(mathematics)"

\end_inset

 ) with
\begin_inset Formula 
\begin{equation}
\nabla\times\v B=(\star(\v d(\v B^{\flat})))^{\sharp}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Part
Draft
\end_layout

\begin_layout Section
Kinetic theory in spacetime
\end_layout

\begin_layout Itemize
Spatial density 
\begin_inset Formula 
\begin{equation}
f(t,\zset)=\frac{1}{NV}\sum_{n}\int_{V}\d^{3}x\,\delta(\zset-\zset_{(n)}(t))
\end{equation}

\end_inset

 becomes 
\begin_inset Quotes eld
\end_inset

event density
\begin_inset Quotes erd
\end_inset

 
\begin_inset Formula 
\begin{equation}
f(t,\zset)=\frac{1}{NV}\int\d^{3}x\,\d t\,\delta(\zset-\zset_{(n)}(s))\delta(t-t_{(n)}(s))
\end{equation}

\end_inset

that looks formally the same, but allows for multiple 
\begin_inset Quotes eld
\end_inset

times
\begin_inset Quotes erd
\end_inset

 for each particle.
\end_layout

\begin_layout Itemize
Particles interact via fields evaluated at 
\begin_inset Formula $F(t_{(n)},\xset_{(n)})$
\end_inset

 and depending on the overall configuration.
\end_layout

\begin_layout Itemize
Collision operators can separate short-scale and long-scale interaction
 by averaging.
\end_layout

\begin_layout Section
Mechanics
\end_layout

\begin_layout Standard
To specify motion of our mechanical system we introduce a 
\begin_inset Formula $4$
\end_inset

-dimensional spacetime manifold 
\begin_inset Formula $Q$
\end_inset

 with coordinates 
\begin_inset Formula $\yset=(x^{0}\equiv t,x^{1},x^{2},x^{3})$
\end_inset

.
 Based on this we define a 
\begin_inset Formula $4N$
\end_inset

-dimensional configuration manifold 
\begin_inset Formula $Q$
\end_inset

 with coordinates 
\begin_inset Formula $\qset=(x_{(n)}^{0}\equiv t_{(n)},x_{(n)}^{1},x_{(n)}^{2},x_{(n)}^{3})$
\end_inset

, meaning 
\begin_inset Quotes eld
\end_inset

particle 
\begin_inset Formula $k$
\end_inset

 was at position 
\begin_inset Formula $x_{(n)}^{1},x_{(n)}^{2},x_{(n)}^{3}$
\end_inset

 at time 
\begin_inset Formula $t_{(n)}$
\end_inset


\begin_inset Quotes erd
\end_inset

.
 The phase-spacetime manifold 
\begin_inset Formula $M$
\end_inset

 follows as the 
\begin_inset Formula $8N$
\end_inset

-dimensional cotangent bundle 
\begin_inset Formula $T^{\star}Q$
\end_inset

 of 
\begin_inset Formula $Q$
\end_inset

.
 According 
\begin_inset Formula $4N$
\end_inset

 canonical momentum coordinates are 
\begin_inset Formula $\pset=(p_{0,\,(n)},p_{1,\,(n)},p_{2,\,(n)},p_{3,\,(n)})$
\end_inset

.
 
\begin_inset Formula 
\begin{equation}
S[\gamma]=\int_{\gamma}\Theta_{L}.\label{eq:S}
\end{equation}

\end_inset


\end_layout

\begin_layout Section
Conservation laws
\end_layout

\begin_layout Itemize
Partial differential equations
\begin_inset Formula 
\begin{equation}
\frac{\partial f}{\partial t}+\nabla_{z}f=s_{f}.
\end{equation}

\end_inset

Write this as
\begin_inset Formula 
\begin{equation}
\partial_{\alpha}f=s_{f}
\end{equation}

\end_inset

with 
\begin_inset Formula $\alpha=0\dots N$
\end_inset

.
\end_layout

\begin_layout Itemize
Ordinary differential equations
\begin_inset Formula 
\begin{equation}
\frac{\d z^{i}}{\d t}=\Lambda^{ik}\partial_{k}H.
\end{equation}

\end_inset

Extended phase-space:
\begin_inset Formula 
\begin{align*}
\frac{\d z^{0}}{\d s} & =\partial_{s}\\
\frac{\d p_{0}}{\d s} & =-\partial_{s}\mathcal{H}
\end{align*}

\end_inset


\end_layout

\begin_layout Itemize
Eulerian
\begin_inset Formula 
\begin{equation}
\frac{\partial a}{\partial t}+\nabla\cdot\Gamma_{a}=s_{a}.
\end{equation}

\end_inset


\end_layout

\begin_layout Itemize
Lagrangian
\begin_inset Formula 
\begin{equation}
\frac{Da}{Dt}=s_{a}.
\end{equation}

\end_inset


\end_layout

\begin_layout Section
Basics
\end_layout

\begin_layout Itemize
What is the use of differential forms?
\end_layout

\begin_deeper
\begin_layout Itemize
We can write even more stuff coordinate-independent than with usual co-contravar
iant notation.
\end_layout

\begin_layout Itemize
The geometric structure of equations and how conservation laws follow become
 apparent.
\end_layout

\begin_layout Itemize
Discrete version tells us how to do numerics the 
\begin_inset Quotes eld
\end_inset

right
\begin_inset Quotes erd
\end_inset

 way.
\end_layout

\begin_layout Itemize
A prime example are Maxwell's equations.
\end_layout

\end_deeper
\begin_layout Itemize
What is a vector?
\end_layout

\begin_deeper
\begin_layout Itemize
It points along a curve as a tangent.
\end_layout

\end_deeper
\begin_layout Itemize
What can we do with a vector?
\end_layout

\begin_deeper
\begin_layout Itemize
We can differentiate along its direction.
\end_layout

\end_deeper
\begin_layout Itemize
What is a covector (1-form)?
\end_layout

\begin_deeper
\begin_layout Itemize
A map that yields a scalar to each vector.
\end_layout

\begin_layout Itemize
A quantity associated with a surface.
\end_layout

\end_deeper
\begin_layout Itemize
What can we do with a covector?
\end_layout

\begin_deeper
\begin_layout Itemize
Apply it to a vector.
 
\end_layout

\begin_layout Itemize
This yields a scalar product i.e.
 the angle between the tangent vector and the surface normal.
\end_layout

\end_deeper
\begin_layout Itemize
What is the metric?
\end_layout

\begin_deeper
\begin_layout Itemize
It defines how lengths and distances work.
\end_layout

\end_deeper
\begin_layout Itemize
How can we use the metric?
\end_layout

\begin_deeper
\begin_layout Itemize
Get the covector to a vector that yields scalar product 1.
\end_layout

\begin_layout Itemize
i.e.
 the plane whose normal vector coincides with a given vector.
\end_layout

\end_deeper
\begin_layout Itemize
What is an inner product?
\end_layout

\begin_deeper
\begin_layout Itemize
Apply 
\begin_inset Formula $p$
\end_inset

-form to vector and get a 
\begin_inset Formula $p-1$
\end_inset

-form.
\end_layout

\begin_layout Itemize
But wait, didn't you say that a 
\begin_inset Formula $p$
\end_inset

-form acts on 
\begin_inset Formula $p$
\end_inset

 vectors?
\end_layout

\begin_layout Itemize
Yes, we define our 
\begin_inset Formula $p-1$
\end_inset

-form such that it gives the same scalar as the 
\begin_inset Formula $p$
\end_inset

-form when the vector is inserted at the first position.
\end_layout

\begin_layout Itemize
Examples:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\v E\cdot\vec{v}=i_{v}\v E$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\vec{v}\times\v B=i_{v}\v B$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\rho\vec{v}=i_{v}\rho$
\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Itemize
What is a Hodge operator?
\end_layout

\begin_deeper
\begin_layout Itemize
Associates a 
\begin_inset Formula $n-p$
\end_inset

-form to a 
\begin_inset Formula $p$
\end_inset

-form.
\end_layout

\end_deeper
\begin_layout Section
How to do a Fourier expansion in curvilinear coordinates?
\end_layout

\begin_layout Subsection
Simple cases
\end_layout

\begin_layout Standard
In Cartesian coordinates the basis 
\begin_inset Formula $\v e_{x},\v e_{y},\v e_{z}$
\end_inset

 doesn't depend on the coordinates and is orthonormal.
 Thus, physical, co- and contravariant components of vector fields become
 one, and we can write any vector field 
\begin_inset Formula $\v B$
\end_inset

 as a Fourier series in either up to all coordinates, i.e.
\begin_inset Formula 
\begin{align}
\v B & =B^{x}(x,y,z)\v e_{x}+B^{y}(x,y,z)\v e_{y}+B^{z}(x,y,z)\v e_{z}\\
 & =\sum_{k=-\infty}^{\infty}\left(B_{k}^{x}(x,y)\v e_{x}+B_{k}^{y}(x,y)\v e_{y}+B_{k}^{z}(x,y)\v e_{z}\right)e^{ikz}\\
 & =\sum_{\v k}(B_{\v k}^{x}\v e_{x}+B_{\v k}^{y}\v e_{y}+B_{\v k}^{z}\v e_{z})e^{i\v k\cdot\v x},
\end{align}

\end_inset

where also in the latter 
\begin_inset Formula $\v k=(k_{x},k_{y},k_{z})$
\end_inset

 all three indexes run from 
\begin_inset Formula $-\infty$
\end_inset

 to 
\begin_inset Formula $\infty$
\end_inset

.
\end_layout

\begin_layout Subsection
General co- and contravariant case
\end_layout

\begin_layout Standard
In any curvilinear coordinates 
\begin_inset Formula $\v x=(x^{1},x^{2},x^{3})$
\end_inset

 we can write a Fourier series for a vector density field 
\begin_inset Formula $B$
\end_inset

 in contravariant density components with
\begin_inset Formula 
\begin{align*}
\v B & =\mathcal{B}^{1}(\v x)\v e_{1}(\v x)+\mathcal{B}^{2}(\v x)\v e_{2}(\v x)+\mathcal{B}^{3}(\v x)\v e_{3}\\
 & =\sum_{n=-\infty}^{\infty}\left(\mathcal{B}_{n}^{1}(x^{1},x^{2})\v e_{1}(\v x)+\mathcal{B}_{n}^{2}(x^{1},x^{2})\v e_{2}(\v x)+\mathcal{B}_{n}^{3}(x^{1},x^{2})\v e_{3}(\v x)\right)e^{inx^{3}}
\end{align*}

\end_inset

or for a vector 
\begin_inset Formula $\v A$
\end_inset

 in covariant components with
\begin_inset Formula 
\begin{align*}
\v A & =A_{1}(\v x)\nabla x^{1}(\v x)+A_{2}(\v x)\nabla x^{2}(\v x)+A_{3}(\v x)\nabla x^{3}(\v x)\\
 & =\sum_{n=-\infty}^{\infty}\left(A_{1n}(x^{1},x^{2})\nabla x^{1}(\v x)+A_{2n}(x^{1},x^{2})\nabla x^{2}(\v x)+A_{3n}(x^{1},x^{2})\nabla x^{3}(\v x)\right)e^{inx^{3}}.
\end{align*}

\end_inset

Here we intentionally keep the full dependency on all coordinates in the
 basis vectors 
\begin_inset Formula $\v e_{k}$
\end_inset

 and 
\begin_inset Formula $\v e^{k}=\nabla x^{k}$
\end_inset

 and apply the Fourier expansion only on vector components.
 It makes a difference whether one expands in co- or contravariant components.
 E.g.
 a vector potential 
\begin_inset Formula $\v A$
\end_inset

 is naturally given in covariant form such that its curl can be nicely separated
 in Fourier form (try it).
\end_layout

\end_body
\end_document