This document provides information on style and notation for "Real Books". It discusses:
- What a Real Book is and the layout of the slides
- How to write and comment on formulae
- Explaining the different parts of individual slides
- Examples of commenting equations term-by-term
- Use of symbols and providing their definitions
- Including references and bibliography
The document aims to establish clear and consistent notation for presenting technical concepts through a series of example slides. It outlines stylistic choices for formatting, commenting equations, and inserting relevant citations and resources.
These slides introduce the estimation of hydraulic conductivity in the case of vadose water. It also discusses a little (very little) of the variability of the hydraulic conductivity across scales.
This is one of the core slides about water in soils and aquifers. It presents Darcy law and its generalisation (Buckingham law) on vadose (unsaturated) case.
This introduce a modern view of the design of urban water management. It promote a design strategy that is aware of all the complexities of the modern urban environment and define where the responsability of a correct management of storm water are
The document discusses The Real Book, which refers to collections of lead sheets that contain standard jazz songs. It provides background on the original Real Book from the 1970s, which was compiled illegally by students at Berklee College of Music. The document then explains that the title "Real Book" is being used for this collection of hydrology lecture slides, which provide systematic knowledge about the topic beyond textbooks. It aims to direct students towards further resources while communicating information.
This document lists 11 corrections to be made in the Real Book. Each correction provides the page number of the song, a description of the error, and the correction that should be made. The corrections range from wrong or missing notes, changes that should be different, and bass figures that need alteration.
These slides introduce the estimation of hydraulic conductivity in the case of vadose water. It also discusses a little (very little) of the variability of the hydraulic conductivity across scales.
This is one of the core slides about water in soils and aquifers. It presents Darcy law and its generalisation (Buckingham law) on vadose (unsaturated) case.
This introduce a modern view of the design of urban water management. It promote a design strategy that is aware of all the complexities of the modern urban environment and define where the responsability of a correct management of storm water are
The document discusses The Real Book, which refers to collections of lead sheets that contain standard jazz songs. It provides background on the original Real Book from the 1970s, which was compiled illegally by students at Berklee College of Music. The document then explains that the title "Real Book" is being used for this collection of hydrology lecture slides, which provide systematic knowledge about the topic beyond textbooks. It aims to direct students towards further resources while communicating information.
This document lists 11 corrections to be made in the Real Book. Each correction provides the page number of the song, a description of the error, and the correction that should be made. The corrections range from wrong or missing notes, changes that should be different, and bass figures that need alteration.
This document discusses the use of satellite soil moisture data for hydrological applications. It summarizes research validating satellite soil moisture products against in situ observations across different scales. It also describes a method called SM2RAIN that estimates rainfall from satellite soil moisture observations by inverting the soil water balance equation. Initial tests of SM2RAIN show good agreement between estimated and observed rainfall.
This contains the lecture about how to read data from the console. And obviously it contains also other information: about UML, about TextIO class and other stuff. See also http://abouthydrology.blogspot.it/2013/07/java-for-hydrologists-101.html for more information and for the other slides
The document discusses a Java program that solves linear equations. It begins by outlining objectives and analyzing the problem of solving for one variable in an equation of the form "ax + b = 0". It then shows the initial coding of a simple program to solve a specific case. The document goes on to discuss improving the program by making it more general and introducing object-oriented programming concepts like classes, methods and information hiding. It provides annotated code and explanations for a class called LinearEquationSolver that takes parameters to solve any linear equation, unless the coefficient of x is 0.
This is the implementation with explanations of a Hello World simple program. It is useful to document keyword and Java modifiers, as well as how to execute a program.
The document provides an introduction to using the Eclipse Java IDE for beginners learning Java. It recommends first understanding basic Java concepts by reading introductory books before using an IDE. It then directs the reader to an external website that provides instructions on installing and using Eclipse's basic features. The document stresses the importance of self-practice and mentions several other tools like Git, Ant, and Maven that programmers should learn but doesn't provide details as the author is also still learning.
The document discusses water and energy budgets. It explains that a budget represents the variation of a given quantity within a control volume over a time interval, and is the algebraic sum of inputs and outputs. It provides examples of water budgets for soil volumes and atmospheric layers, accounting for precipitation, evapotranspiration, runoff and other fluxes. It also discusses the components of an energy budget, including net radiation, heat conduction, heat of vaporization and more.
The document summarizes the activities of the Platform Water Management in the Alps over the past two years and outlines its planned activities for the next period. It discusses workshops held on sediment management, hydropeaking, and hydropower that brought together administrators, practitioners, and stakeholders. It also describes dissemination of guidelines on small hydropower and platform meetings. Going forward, the document outlines three planned workshops on local adaptation to climate change, flood risk prevention, and river management, as well as a conference on water in the Alps. The goals are to address EU directives in an alpine context and local adaptation to climate change.
The document provides an introduction to hydrology, including:
- Defining hydrology as the science studying the water cycle and flows between the atmosphere, land, and oceans.
- Describing the key elements of the water cycle, including precipitation, infiltration, evaporation, and the spatial and temporal scales involved.
- Noting that the water cycle sustains life on Earth, shapes its surface, and regulates the climate.
This document provides an introduction to solar radiation and its role in powering the water cycle. It discusses the composition and structure of the Sun, and how it produces radiation through nuclear fusion. While solar radiation is generally constant, it exhibits variations in the form of solar spots and an 11-year activity cycle. The amount of radiation emitted by any body is determined by the Stefan-Boltzmann law, which relates radiation to the body's temperature and emissivity.
The document discusses the measurement and representation of hydrological quantities. It notes that hydrological data has complex trends that are nonlinear and influenced by many factors. Statistical tools must be used to describe hydrological quantities given their spatiotemporal variability. Examples of typical problems in measuring quantities like precipitation, river flows, and soil moisture are provided.
The document discusses various hydrological measurement quantities and instruments. It describes 8 main hydrological quantities of interest: temperature, humidity, precipitation, radiation, wind, pressure, wetting, and evapotranspiration. It then explains principles and instruments for measuring temperature, humidity, and soil moisture, including thermometers, hygrometers, psychrometers, lysimeters, tensiometers, and instruments measuring electrical conductivity, thermal conductivity, and dielectric constants.
This document discusses peak river flows and flow hydrology. It introduces the concept of a peak flow, shows a graph of discharge over time as an example, and discusses precipitation patterns and the calculation of effective precipitation. It also discusses the instantaneous unit hydrograph method for summing surface runoff over a basin to determine discharge at the basin outlet.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
This document discusses the use of satellite soil moisture data for hydrological applications. It summarizes research validating satellite soil moisture products against in situ observations across different scales. It also describes a method called SM2RAIN that estimates rainfall from satellite soil moisture observations by inverting the soil water balance equation. Initial tests of SM2RAIN show good agreement between estimated and observed rainfall.
This contains the lecture about how to read data from the console. And obviously it contains also other information: about UML, about TextIO class and other stuff. See also http://abouthydrology.blogspot.it/2013/07/java-for-hydrologists-101.html for more information and for the other slides
The document discusses a Java program that solves linear equations. It begins by outlining objectives and analyzing the problem of solving for one variable in an equation of the form "ax + b = 0". It then shows the initial coding of a simple program to solve a specific case. The document goes on to discuss improving the program by making it more general and introducing object-oriented programming concepts like classes, methods and information hiding. It provides annotated code and explanations for a class called LinearEquationSolver that takes parameters to solve any linear equation, unless the coefficient of x is 0.
This is the implementation with explanations of a Hello World simple program. It is useful to document keyword and Java modifiers, as well as how to execute a program.
The document provides an introduction to using the Eclipse Java IDE for beginners learning Java. It recommends first understanding basic Java concepts by reading introductory books before using an IDE. It then directs the reader to an external website that provides instructions on installing and using Eclipse's basic features. The document stresses the importance of self-practice and mentions several other tools like Git, Ant, and Maven that programmers should learn but doesn't provide details as the author is also still learning.
The document discusses water and energy budgets. It explains that a budget represents the variation of a given quantity within a control volume over a time interval, and is the algebraic sum of inputs and outputs. It provides examples of water budgets for soil volumes and atmospheric layers, accounting for precipitation, evapotranspiration, runoff and other fluxes. It also discusses the components of an energy budget, including net radiation, heat conduction, heat of vaporization and more.
The document summarizes the activities of the Platform Water Management in the Alps over the past two years and outlines its planned activities for the next period. It discusses workshops held on sediment management, hydropeaking, and hydropower that brought together administrators, practitioners, and stakeholders. It also describes dissemination of guidelines on small hydropower and platform meetings. Going forward, the document outlines three planned workshops on local adaptation to climate change, flood risk prevention, and river management, as well as a conference on water in the Alps. The goals are to address EU directives in an alpine context and local adaptation to climate change.
The document provides an introduction to hydrology, including:
- Defining hydrology as the science studying the water cycle and flows between the atmosphere, land, and oceans.
- Describing the key elements of the water cycle, including precipitation, infiltration, evaporation, and the spatial and temporal scales involved.
- Noting that the water cycle sustains life on Earth, shapes its surface, and regulates the climate.
This document provides an introduction to solar radiation and its role in powering the water cycle. It discusses the composition and structure of the Sun, and how it produces radiation through nuclear fusion. While solar radiation is generally constant, it exhibits variations in the form of solar spots and an 11-year activity cycle. The amount of radiation emitted by any body is determined by the Stefan-Boltzmann law, which relates radiation to the body's temperature and emissivity.
The document discusses the measurement and representation of hydrological quantities. It notes that hydrological data has complex trends that are nonlinear and influenced by many factors. Statistical tools must be used to describe hydrological quantities given their spatiotemporal variability. Examples of typical problems in measuring quantities like precipitation, river flows, and soil moisture are provided.
The document discusses various hydrological measurement quantities and instruments. It describes 8 main hydrological quantities of interest: temperature, humidity, precipitation, radiation, wind, pressure, wetting, and evapotranspiration. It then explains principles and instruments for measuring temperature, humidity, and soil moisture, including thermometers, hygrometers, psychrometers, lysimeters, tensiometers, and instruments measuring electrical conductivity, thermal conductivity, and dielectric constants.
This document discusses peak river flows and flow hydrology. It introduces the concept of a peak flow, shows a graph of discharge over time as an example, and discusses precipitation patterns and the calculation of effective precipitation. It also discusses the instantaneous unit hydrograph method for summing surface runoff over a basin to determine discharge at the basin outlet.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
0-RealBookStyleAndNotation
1. The Real Books:
On Style and Notation
R. Rigon- Il tavolo di lavoro di Remo wolf
Riccardo Rigon
Tuesday, February 26, 13
2. “Standards are nice if each
one of us has his own”
Sandro Marani
Tuesday, February 26, 13
3. 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
3
R. Rigon
Tuesday, February 26, 13
4. 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
4
R. Rigon
Tuesday, February 26, 13
5. 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
5
R. Rigon
Tuesday, February 26, 13
6. 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.
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.
Change of mass of the
snow in the control
volume per unit time
6
R. Rigon
Tuesday, February 26, 13
7. Notation
Notes on Style: an example 2/3
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 H f
=P
dt f
Total precipitation
7
R. Rigon
Tuesday, February 26, 13
8. Notation
Notes on Style: an example 3/3
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.
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.
Heating of snow
divided by the enthalpy
of fusion of ice
8
R. Rigon
Tuesday, February 26, 13
9. 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)
9
R. Rigon
Tuesday, February 26, 13
10. 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.
10
R. Rigon
Tuesday, February 26, 13
11. Notation
Symbols
Where possible, there will be one or more tables listing the symbols
used, like the one below:
11
R. Rigon
Tuesday, February 26, 13
12. Symbols
The aim, wherever possible, is to use standard symbols that are
different for different quantities.
12
R. Rigon
Tuesday, February 26, 13
13. Symbols
The Name is as in the CF Conventions (http.CF), or is given in that
style
13
R. Rigon
Tuesday, February 26, 13
14. Symbols
The unit of measure
should always be shown
14
R. Rigon
Tuesday, February 26, 13
15. 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/
15
R. Rigon
Tuesday, February 26, 13
16. 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,
•..........
16
R. Rigon
Tuesday, February 26, 13
17. Basic Notation for Scalar, Vector
and Tensor Fields, and Matrices
Bruno Munari - Libri illeggibili
Tuesday, February 26, 13
18. 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.
18
R. Rigon
Tuesday, February 26, 13
19. 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
19
R. Rigon
Tuesday, February 26, 13
20. 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 }
20
R. Rigon
Tuesday, February 26, 13
21. 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.
21
R. Rigon
Tuesday, February 26, 13
22. 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.
22
R. Rigon
Tuesday, February 26, 13
23. 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.
23
R. Rigon
Tuesday, February 26, 13
24. 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
24
R. Rigon
Tuesday, February 26, 13
25. 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
e m p t y
space
25
R. Rigon
Tuesday, February 26, 13
26. 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
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.
26
R. Rigon
Tuesday, February 26, 13
27. 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
temporal
i n d e x ,
preceded by
a comma
27
R. Rigon
Tuesday, February 26, 13
28. 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
iterative index,
preceded by a
semicolon
28
R. Rigon
Tuesday, February 26, 13
29. Discrete Representation
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
29
R. Rigon
Tuesday, February 26, 13
30. Discrete Representation
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
30
R. Rigon
Tuesday, February 26, 13
31. Discrete Representation
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:
31
R. Rigon
Tuesday, February 26, 13
32. Discrete Representation
where the symbol is used to identify a column vector
32
R. Rigon
Tuesday, February 26, 13
33. Discrete Representation
where the symbol is used to identify a row type of vector
33
R. Rigon
Tuesday, February 26, 13
34. Discrete Representation
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.
34
R. Rigon
Tuesday, February 26, 13
35. Discrete Representation
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
35
R. Rigon
Tuesday, February 26, 13
36. Discrete Representation
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)
36
R. Rigon
Tuesday, February 26, 13
37. Discrete Representation
Cell points and face points in a structured grid:
37
R. Rigon
Tuesday, February 26, 13
38. Discrete Representation
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
38
R. Rigon
Tuesday, February 26, 13
39. 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 }
39
R. Rigon
Tuesday, February 26, 13
40. 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.
40
R. Rigon
Tuesday, February 26, 13
41. 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
41
R. Rigon
Tuesday, February 26, 13
42. 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).
42
R. Rigon
Tuesday, February 26, 13
43. 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.
43
R. Rigon
Tuesday, February 26, 13
44. 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.
44
R. Rigon
Tuesday, February 26, 13
45. Thank you for your attention.
G.Ulrici, 2000 ?
45
R. Rigon
Tuesday, February 26, 13