0-realbookstyleandnotation

Riccardo Rigon

The Real Books:On Style and Notation

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Tuesday, February 26, 13

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

Sandro Marani


R. Rigon

3

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

Notation


R. Rigon

4

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�dt

= P � �H�f

�f

Notation


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5

dM�dt

= P � �H�f

�f

Conservation of mass of snow

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.

Notation

The three slides show how to

comment an equation, term by

term. Slowness is necessary to

reproduces some optimal flux

of information.


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6

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.

dM�dt

= P � �H�f

�f

Change of mass of the snow in the control volume per unit time

Notation





of information.


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step.

dM�dt

= P � �H�f

�f

Total precipitation


Notation


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step.

dM�dt

= P � �H�f

�f

H e a t i n g o f s n o w divided by the enthalpy of fusion of ice


Notation





of information.


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The slides have some standard information: a general index

The slides have some standard information: the authors of the contribution

The slide number: gives the audience a reference point

For these slides a Creative Commons License has been used (http.cc)

Notes on Style:

The centre of the

slide is white: this is

f o r i m p r o v e d

visibility and to avoid

w a s t a g e o f t o n e r

when printing. The

cover slide, on the

other hand, is all blue

with an image.

The slides have some standard information: authors

NotationR

igon

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13


http://www.creativecommons.org

http://www.creativecommons.org

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10

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.

Notation


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Symbols

Where possible, there will be one or more tables listing the symbols

used, like the one below:

Notation


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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


http://cf-pcmdi.llnl.gov


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SymbolsThe 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/


http://en.wikipedia.org/wiki/Real_Book

http://en.wikipedia.org/wiki/Real_Book

http://creatie

http://creatie



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Bibliography

Each set of these slides contains a bibliography.

•S. Swallow - The Real Book CD,

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

•..........


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

Bru

no M

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ari

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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.


R. Rigon

Let Ulw be a space-time field. Then

Ulw(⌥x, t) = Ulw(x, y, z, t)

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

is a vector field. Other notations for vectors are possible, but not used.

⌥Ulw(⌥x, t) = ⌥Ulw(x, y, z, t)

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

Basic Basics

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R. Rigon

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

The components of the vector field can be written as:

or, by omitting the dependence on the space-time variables, as:

⌥Ulw(⌥x, 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:

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

Basic Basics

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R. Rigon

The normal derivative of the field with respect to the variable x can be

expressed in the canonical form:

d

dx�Ulw(�x, t) =

d

dx�Ulw(x, y, z, t) =

�d

dxUlw(�x, t)x,

d

dxUlw(�x, t)y,

d

dxUlw(�x, t)z

⇥

⇥x�Ulw(�x, t) = ⇥x

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

The partial derivative of the field with respect to the variable x can also be

expressed as:

The partial derivative of the field with respect to the variable x can also be expressed in the canonical form:

Other forms are possible but not used.

⇥

⇥x�Ulw(�x, t) =

⇥

⇥x�Ulw(x, y, z, t) =

�⇥

⇥xUlw(�x, t)x,

⇥

⇥xUlw(�x, t)y,

⇥

⇥xUlw(�x, t)z

⇥

Derivatives

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R. Rigon

Gradient and DivergenceThe gradient of a scalar field is expressed, in the canonical form, as:

⌃⇤Ulw(⌃x, t) = {⇥xUlw(⌃x, t), ⇥yUlw(⌃x, t), ⇥zUlw(⌃x, t)}

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

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.

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

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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:

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.

i � {x, y, x}

⇥ · ⌃Ulw(⌃x, t) = ⇥iUlw(⌃x, t)i = ⇥iUlw(⌃x, t)i

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R. Rigon

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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R. Rigon



Ulw ij,t;k

e m p t y space


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R. Rigon



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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R. Rigon



Ulw ij,t;k

t e m p o r a l i n d e x , preceded by a comma

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R. Rigon



Ulw ij,t;k

iterative index, preceded by a semicolon


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R. Rigon

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 omitted29



R. Rigon

Possible alternatives with the same meaning are:

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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.


R. Rigon

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 ( ):


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As in the previous cases the comma can be omitted


R. Rigon

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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R. Rigon

Cell points and face points in a structured grid:

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R. Rigon

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.

U ,tlw

means the field Ulw at a generic cell at time t


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R. Rigon

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

Discrete Representation of Vector Components

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

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R. Rigon

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

Tensors

Ulw(⌃x, t) = Ulw(x, y, z, t)

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

�

⇤Ulw(⇧x, t)xx Ulw(⇧x, t)xy Ulw(⇧x, t)xz

Ulw(⇧x, t)yx Ulw(⇧x, t)yy Ulw(⇧x, t)yz

Ulw(⇧x, t)zx Ulw(⇧x, t)zy Ulw(⇧x, t)zz

⇥

⌅

The components are not written with bold characters.

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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


R. Rigon

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

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).

Tensors are matrices, and matrix notation applies to tensors

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R. Rigon

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

Tensors are matrices, and matrix notation applies to tensors

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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When doing thermodynamics

U = U(S, V, Mw)

thus, its differential is:

Internal energy can be written, for instance, as :

dU( ) = T ( )dS � p( )dV + µw( )dMw

where , and 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.

µw( )T ( ) p( ) ( )

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.


R. Rigon

Thank you for your attention.

G.U

lric

i, 2

00

0 ?

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