0-RealBookStyleAndNotation

The Real Books:
On Style and Notation
R. Rigon- Il tavolo di lavoro di Remo wolf

Riccardo Rigon

Tuesday, February 26, 13

“Standards are nice if each
one of us has his own”
Sandro Marani


Notation

Objectives
Each set of these slides contains a summary, or
description, of the communication objectives that

want to be achieved.

•These slides will explain what a Real Book is

•The layout of these slides is explained

•They will explain how to write and comment the formulae

•The various parts of the single slides are also explained

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Notation

Notes on Style

For these slides I have chosen to use the Lucida Bright font, at 24 point size,
with justified text. The titles have been centred and they have been written in a
36 point Lucida Bright font.
The notes are in 18 point Lucida Bright. The references are in 14 point Lucida
Bright.

The choice of font is linked to the formulae, which are pdf images created
with LaTeX (specifically LaTeXit! for Mac), using the Computer Modern font,
which is very similar to Lucida Bright. The formulae usually use a 36 point font
size. There follows an example.

dM H f
=P
dt f

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Notation

Notes on Style

Experience teaches that, in order to reproduce the communicative effect of
writing by hand on a blackboard, the formulae need to commented. For these
slides I have chosen the following method: the formula is “boxed” in red (2 pt)
and a red arrow points to an explanation in italics.

The three slides show how to
dM H comment an equation, term by
f
=P term. Slowness is necessary to
dt f reproduces some optimal flux
of information.

Conservation of mass of snow

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Notation

Notes on Style: an example 1/3

The evolution of the water equivalent of snow is found by solving the mass balance
equation: the liquid and solid precipitation less the flow of water due to melting
and sublimation is equal to the variation in the water equivalent during the time
step.

f
of information.
Change of mass of the
snow in the control
volume per unit time
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Notation


step.

dM H f
=P
dt f

Total precipitation

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Notation


step.

f
of information.

Heating of snow
divided by the enthalpy
of fusion of ice
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Notation

Notes on Style:
The slides have some The centre of the
standard information:
a general index slide is white: this is
for improved
visibility and to avoid
wastage of toner
The slides have some when printing. The
The slide number:
standard information: cover slide, on the
gives the audience a
authors other hand, is all blue reference point
with an image.
Rigon, 2013

The slides have some
For these slides a
standard information:
Creative Commons
the authors of the
License has been
contribution
used (http.cc)

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Notation

Other Notes:

The formulae have been written using LaTeXit, and they are alive, in the
sense that dragging them back to LaTeXit, the code that generated them
reappears.

Generally, wherever possible, parts of the calculation code or graphic
generation code are also given.

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Notation

Symbols
Where possible, there will be one or more tables listing the symbols
used, like the one below:

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Symbols

The aim, wherever possible, is to use standard symbols that are
different for different quantities.

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Symbols

The Name is as in the CF Conventions (http.CF), or is given in that
style

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Symbols
The unit of measure
should always be shown

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Risorse web

•http.wp - http://en.wikipedia.org/wiki/Real_Book - Last accessed May, 7, 2009
•http.cc - http://creative.commons.org - Last accessed May, 7, 2009

•http.CF -http://cf-pcmdi.llnl.gov/

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Bibliography

Each set of these slides contains a bibliography.

•R. Rosso, Corso di Infrastrutture Idrauliche, Sistemi di drenaggio urbano, The
Real book, CUSL, 2002

•S. Swallow - The Real Book CD,
•..........

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Basic Notation for Scalar, Vector
and Tensor Fields, and Matrices

Bruno Munari - Libri illeggibili


Objectives

•In these slides the notational rules used in the Real Books are defined.

•In particular, explanation is given on how to write the formulae so that the
indices and various graphic aspects can be interpreted univocally.

•However these are guidelines that can be violated in practical cases in favor
of simplicity of notation.

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Basic Basics
Let Ulw be a space-time field. Then

Ulw (⌥ , t) = Ulw (x, y, z, t)
x
is a scalar field. The field can be independent of some space variable or
time, which is then omitted. Whether the vector is 2-D or 3-D depends
on the context. On the other hand

⌥ x ⌥
Ulw (⌥ , t) = Ulw (x, y, z, t)
is a vector field. Other notations for vectors are possible, but not used.

⌥ x ⌥
Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (⌥ , t)x , Ulw (⌥ , t)y , Ulw (⌥ , t)z }
x x x
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Basic Basics

The components of the vector field can be written as:

⌥ x ⌥
Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (⌥ , t)x , Ulw (⌥ , t)y , Ulw (⌥ , t)z }
x x x
or, by omitting the dependence on the space-time variables, as:

⌥ x ⌥
Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw x , Ulw y , Ulw z }

Please take note of the space between the “lw” and coordinate index.
Sometimes just the space variable, or the time variable, dependence can be
omitted to simplify the notation as:

⌥ x ⌥
Ulw (⌥ , t) = Ulw (x, y, z, t) = {Ulw (t)x , Ulw (t)y , Ulw (t)z }

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Derivatives
The normal derivative of the field with respect to the variable x can be
expressed in the canonical form:
⇥
d d d d d
Ulw (x, t) = Ulw (x, y, z, t) = Ulw (x, t)x , Ulw (x, t)y , Ulw (x, t)z
dx dx dx dx dx

The partial derivative of the field with respect to the variable x can also be
expressed in the canonical form:
⇥
⇥ ⇥ ⇥ ⇥ ⇥
Ulw (x, t) = Ulw (x, y, z, t) = Ulw (x, t)x , Ulw (x, t)y , Ulw (x, t)z
⇥x ⇥x ⇥x ⇥x ⇥x
The partial derivative of the field with respect to the variable x can also be
expressed as:

⇥x Ulw (x, t) = ⇥x Ulw (x, y, z, t) = {⇥x Ulw (x, t)x , ⇥x Ulw (x, t)y , ⇥x Ulw (x, t)z }

Other forms are possible but not used.
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Gradient and Divergence
The gradient of a scalar field is expressed, in the canonical form, as:

⌃
⇤Ulw (⌃ , t) = {⇥x Ulw (⌃ , t), ⇥y Ulw (⌃ , t), ⇥z Ulw (⌃ , t)}
x x x x

The divergence of a vector field is expressed, in the canonical form, as:

⌃ x
⇥ · Ulw (⌃ , t) = ⇥x Ulw (⌃ , t)x + ⇥y Ulw (⌃ , t)y + ⇥z Ulw (⌃ , t)z
x x x

where on the left there is the geometric (coordinate independent) form,
and on the right are the gradients in Cartesian coordinates. Vector
symbol above the divergence is omitted to remind that the result of the
application of the operator to a vector is a scalar.

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R. Rigon

Gradient and Divergence

The divergence can also be expressed in a more compact form using
the Einstein summation convention:

⌃ x
⇥ · Ulw (⌃ , t) = ⇥ i Ulw (⌃ , t)i = ⇥i Ulw (⌃ , t)i
x x
i {x, y, x}

meaning that when an index variable appears twice in a single term,
once in an upper (superscript) and once in a lower (subscript) position,
there is a summation over all of its possible values.

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Discrete Representation

It is interesting to see how scalar and vector fields are represented
when they are discretised into a grid

Ulw ij,t;k

subscript
symbol

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Ulw ij,t;k

e m p t y
space

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Ulw ij,t;k

spatial index, first index
refers to the cell (center) the
second to the cell face, which
is then j(i). If only one index is
present it is a cell index.

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Ulw ij,t;k

temporal
i n d e x ,
preceded by
a comma

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Ulw ij,t;k

iterative index,
preceded by a
semicolon

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Possible alternatives with the same meaning are:

Subscripts and superscripts can be omitted, for simplicity, when the
meaning of the variable is clear from the context. All of the above are
calculated at/across face j of cell i at time step t and it is iteration k.
When there is no ambiguity, also the comma can be omitted
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Possible alternatives with the same meaning are:

All the above quantities are calculated for cell i at time step t and it is
iteration k

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When a single index is presented, it can be, for instance

with varying i. Therefore, a “vector”, meaning an array of data, can be built:

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where the symbol is used to identify a column vector

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where the symbol is used to identify a row type of vector

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The two symbols

or “harpoon” are used for distinguishing this type of vector from the spatial
euclidean vectors that have certain particular transformation rules upon
rotations in space.

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If the cell in which the system is discretized is a square in a structured
cartesian grid, then the same as above applies, but the cell is identified by
the row and column numbers enclosed in ( ):

As in the previous cases the comma can be omitted

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If the cell is a square in a structured cartesian grid, then the same as
above applies, but the cell face is identified by the row and column
numbers enclosed in ( ) with +1/2 (or -1/2)

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Cell points and face points in a structured grid:

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If position or time or iteration are identifiable from the context, or they
are unimportant or a non-applicable feature, then they can be omitted

means the field Ulw at the face between position i,j and i,j+1 in a
cartesian grid at a known time.

Ulw i

means the field Ulw at cell i in an unstructured grid at a known or
unspecified time.

,t
Ulw
means the field Ulw at a generic cell at time t

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Discrete Representation of Vector Components

These are represented with a straightforward extension of what was
used with scalars:

⇤
Ulw ij,t;k = {Ulw.x ij,t;k , Ulw.y ij,t;k , Ulw.z ij,t;k }

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Tensors
A tensors field is represented by bold letters (either lower or upper case)

Ulw (⌃ , t) = Ulw (x, y, z, t)
x

In this case Ulw is a 3 x 3 tensor field with components:

⇥
Ulw (⇧ , t)xx
x Ulw (⇧ , t)xy
x Ulw (⇧ , t)xz
x
⇤ Ulw (⇧ , t)yx
x Ulw (⇧ , t)yy
x Ulw (⇧ , t)yz ⌅
x
Ulw (⇧ , t)zx
x Ulw (⇧ , t)zy
x Ulw (⇧ , t)zz
x

The components are not written with bold characters.
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R. Rigon

Tensors
However, a tensor by components representation is preferable. So U
becomes:

Or, when the notation is not ambiguous (not to be confounded with the (ij)
element of a grid) simply:

The context says if the subscripts refer to a grid point or to the component
of a tensors. This is deemed necessary to avoid extra

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R. Rigon

Tensors

All the rules given for scalars and vectors apply consistently to tensors

Tensors are matrices, and matrix notation
applies to tensors

However, bear in mind that scalars, vectors, and tensors are geometric
objects which have properties that are independent of the choice of
reference system (i.e. independent of the origin, the base, and the
orientation of the space-time vector space) and the coordinate system
(i.e. cartesian, cylindrical or curvilinear or other).

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Tensors are matrices, and matrix notation
applies to tensors

Thus, while tensor indices always refers to space-time, matrix indices do
not.

Remember also that divergence, gradient and curl are themselves
geometric objects and obey the same rules as tensors. By changing
coordinate system, they change their components but not their
geometric properties.

These geometric properties, in fact, should be preserved in a proper
discretisation, since they are intimately related to the Conservation Laws
of Physics.

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R. Rigon

When doing thermodynamics

Internal energy can be written, for instance, as :

U = U (S, V, Mw )
thus, its differential is:

dU ( ) = T ( )dS p( )dV + µw ( )dMw
where T ( ) , p( ) and µw ( ) are followed by ( ) to indicate that they are
functions and not independent variables. Usually they are also functions of space
and time (fields), but this dependence remains implicit.

This notation is convenient since the real dependence of each function on the
variables S, V, Mw depends on the system under analysis, and is unspecified a-priori.
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R. Rigon

Thank you for your attention.

G.Ulrici, 2000 ?

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R. Rigon

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