0.c basic notations

Basic Notation for scalar, vector,
Tensor fields and Matrixes

Bruno Munari - Libri illeggibili

Riccardo Rigon

Thursday, September 2, 2010

“Gli standard sono belli se
ognuno ha il suo”
Sandro Marani


The Real Books

Obbiettivi

•In queste Slides si definiscono delle regole per la notazione usate nelle slides che
seguono.

•In particolare si spiega come scrivere le formule in modo che il significato
di indici e vari aspetti grafici della scrittura siano interpretati in modo
univoco.

3

Riccardo Rigon


Basic Notation

Basics of Basics
Let Ulw be a spatio -temporal field. Then

Ulw ( , t) = Ulw (x, y, z, t)
x

is a scalar field. The field can be independent of some space
variabile ot the time, which is then omitted, if the vector is 2-
D or 3-D depends on hte context. Instead
x
Ulw ( , t) = Ulw (x, y, z, t)

is a vector field. Other notation for vector 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
4

Riccardo Rigon


Basic Notation

Basics of Basics
The components of the vector field can be written according
to:
x
Ulw ( , t) = Ulw (x, y, z, t) = {Ulw ( , t)x , Ulw ( , t)y , Ulw ( , t)z }
x x x

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

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

Please notice the space between the “lw” and coordinate
index. Sometimes just the space variabile 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 }

5

Riccardo Rigon


Basic Notation

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 ( , t) =
x Ulw (x, y, z, t) = Ulw ( , t)x , Ulw ( , t)y , Ulw ( , t)z
x x x
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 ( , t) =
x Ulw (x, y, z, t) = Ulw ( , t)x ,
x Ulw ( , t)y ,
x Ulw ( , t)z
x
∂x ∂x ∂x ∂x ∂x

The partial derivative of the field with respect to the variable
x but also as
x
∂x Ulw ( , t) = ∂x Ulw (x, y, z, t) = {∂x Ulw ( , t)x , ∂x Ulw ( , t)y , ∂x Ulw ( , t)z }
x x x

Other forms are possible but not used
6

Riccardo Rigon


Basic Notation

Gradient and Divergence
The gradient of a scalar field is expressed, in the canonical
form, or 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, or 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, there is the gradients
in Cartesian coordinates.
7

Riccardo Rigon


Basic Notation

Gradient and Divergence
The divergence can be expressed also in a more compact
form using the Einstein’s convention
lw ( , t) = ∂ i Ulw ( , t)i = ∂i Ulw ( , t)i
∇·U x x x
i ∈ {x, y, x}

meaning that, when double indexing is up and down there is
a summation which spans all the values of the sub(super)-
script

8

Riccardo Rigon


Basic Notation

Discrete representation
It is interesting to see how scalar and vector field are
represented when they are discretized on a grid

Ulw ij,t;k

subscript
symbol

9

Riccardo Rigon


Basic Notation


Ulw ij,t;k

empty
space

10

Riccardo Rigon


Basic Notation


Ulw ij,t;k

spatial index, first index
refers to the cell (center)
the second to the cell
face, which is j(i) then. If
only one index is present
it is a cell index. 11

Riccardo Rigon


Basic Notation


Ulw ij,t;k

temporal
i n d e x ,
preceded
by a
comma

12

Riccardo Rigon


Basic Notation


Ulw ij,t;k

iterative
i n d e x ,
preceded
by a
semicolon

13

Riccardo Rigon


Basic Notation

Possible alternative with the same meaning

Ulw ij,t;k
,t
Ulw ij;k
,t;k
Ulw ij
;k
Ulw ij,t
U ij,t;k

subscripts or superscripts can be omitted for simplicity
when the meaning of the variable is clear from the context.
All the above are calculated for (across) the face j of the cell i
at time step t and it is iteration k 14

Riccardo Rigon


Basic Notation

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

Ulw i,t;k
,t
Ulw i;k
,t;k
Ulw i
;k
Ulw i,t
U i,t;k

15

Riccardo Rigon


Basic Notation

If the cell is a square in a structured cartesian grid, then the
same as above applies but the cell is identified by the row and
colums number enclosed by ( )
Ulw (i,j),t;k
,t
Ulw (i,j);k
,t;k
Ulw (i,j)
;k
Ulw (i,j),t
U (i,j),t;k

16

Riccardo Rigon


Basic Notation

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 colums number enclosed by ( ) with +1/2 (or -1/2)

Ulw (i,j+1/2),t;k
,t
Ulw (i,j+1/2);k
,t;k
Ulw (i,j+1/2)
;k
Ulw (i,j+1/2),t
U (i,j+1/2),t;k

17

Riccardo Rigon


Basic Notation

Cell points and face points in a structured grid

18

Riccardo Rigon


Basic Notation

If position or time, or iteration are known from the context,
or unimportant or a non applicable feature can be omitted

Ulw (i,j+1/2)

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

Means the field Ulw at cell i in an unstructured grid at known
or unspecified time
,t
Ulw

Means the field Ulw at generic cell at time t
19

Riccardo Rigon


Basic Notation

Discrete representation of vector
components

Are built upon a straightforward extension of what made
with scalars

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

20

Riccardo Rigon


Basic Notation

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

Components use a non bold character.
21

Riccardo Rigon


Basic Notation

Tensors

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

Tensors are matrixes, and matrixes notation
follows the same rules of tensors

However remind that scalar, vector and tensors are geometric objects
which have properties which are independent from the choice of any
reference system (i.e. independent from the origin, base, and orientation
of the space-time vector space) and coordinate system (i.e. cartesian,
cylindrical or curvilinear or other).

22

Riccardo Rigon


Basic Notation

Tensors are matrixes, and matrixes notation
follows the same rules of tensors

Thus, while tensors’ indexes refers always to space-time, matrixes
indexes do not.

Remind also that divergence, gradient and curl are themselves geometric
objects and obey the same rules than tensors. With changing coordinate,
they change their components but not their geometric properties.

This geometric properties in fact should be preserved by proper a
discretization, since they are intimately related to the conservation laws
of Physics.

23

Riccardo Rigon


Basic Notation

G. Ulrici -

24

Riccardo Rigon


0.c basic notations

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (20)

Similar a 0.c basic notations

Similar a 0.c basic notations (20)

Más de Riccardo Rigon

Más de Riccardo Rigon (20)

Último

Último (20)

0.c basic notations