2. The classification of physical quantities
Physical
quantities
are
classified
according
to
several
criteria:
1. In direction. The physical quantity that reflects the direction of motion, called vector
quantity,
otherwise
scalar
quantity.
2. By the character of dimension. The physical quantity which has dimension formula,
in which at least one dimension of a non-zero exponent, called dimensional quantity. If
all dimensions have a zero exponent, then this quantity called dimensionless quantity.
International Vocabulary of Metrology JCGM 200:2012 allows the use by the term
"dimensionless quantity" only for historical reasons, but recommended to prefer the term
"quantity
of
dimension
one".
3. If possible summation. The physical quantity is called the additive quantity, if its
values ​can be summarized, multiplied by a numerical factor, divided against each other,
as, for example, that can be done with the force or moment of force, and non-additive
quantity, if the mathematical operations have no physical meaning, such as a
thermodynamic temperature whose value does not make sense to add or subtract.
4. With respect to the of a physical quantity physical system. The physical quantity is
called extensive quantity, if its value is the sum of the values ​of the same physical
quantities for subsystems of which comprises a system such as that of volume and
intensive quantity, if its value does not depend on the size of the system, such as in
thermodynamic temperature.
Carnap Criteria
Rudolf Carnap (1891 – 1970)
German-born philosopher
For any quantity, a competent metrologist
with suitable equipment is one who has a realization
of four axiom:
1. Within the domain of objects accessible one object must be the unit
2. Within the domain of objects accessible one object must be zero
3. There must be a realizable operation to order the objects as to magnitude or intensity of the
measured quantity.
4. There must be an algorithm to generate a scale between zero and the unit
To make the above clear, consider the following system:
• The quantity of interest, temperature;
• the object, a human body;
• the unit, the boiling point of water 100°;
• the zero, the triple point of water 0°;
• the ordering operator, the height of a mercury column in a uniform bore tubing connected
to a reservoir capable of being placed in a body cavity;
• the scale, shall be a linear subdivision between the heights of the column when in
equilibrium with boiling water and water at the triple point
Temperature:
linear scale
But pH scale is logarithmic!
Slide 5 of 19
• Math and science were invented by humans to describe
and understand the world around us. We live in a (at
least) four-dimensional world governed by the passing of
time and three space dimensions; up and down, left and
right, and back and forth. We observe that there are
some quantities and processes in our world that depend
on the direction in which they occur, and there are
some quantities that do not depend on direction. For
example, the volume of an object, the three-dimensional
space that an object occupies, does not depend on
direction.
Frame of reference
Motion (or other property) of a body is always described with reference to some well
defined coordinate system. This coordinate system is referred to as 'frame of reference'.
In three dimensional space a frame of reference consists of three mutually perpendicular
lines called 'axes of frame of reference' meeting at a single point or origin. The
coordinates of the origin are O(0,0,0) and that of any other point 'P' in space are
P(x,y,z).The line joining the points O and P is called the position vector of the point P
with respect to O.
7
Frame of reference example
Frame of Reference
Transformations
Translation
Rotation
Reflection
Tensor
Any tensor with respect to a basis is represented
by a multidimensional array. For example, a
linear operator is represented in a basis as a
two-dimensional square n × n array. The
numbers in the multidimensional array are known
as the scalar components of the tensor or simply
its components. They are denoted by indices
giving their position in the array, as subscripts
and superscripts, following the symbolic name of
the tensor. For example, the components of an
order 2 tensor T could be denoted Tij , where i
and j are indices running from 1 to n, or also by
Tij. Whether an index is displayed as a
superscript or subscript depends on the
transformation properties of the tensor,
described below. The total number of indices
required to identify each component uniquely is
equal to the dimension of the array, and is called
the order, degree or rank of the tensor. However,
the term "rank" generally has another meaning in
the context of matrices and tensors.
Properties of scalar, vector and tensor quantities for coordinate
transformations
Physical
quantity
Translation
Rotation
Reflection
Example
Scalar
invariant
invariant
invariant
Mass ,
temperature
Vector
invariant
Components
change
One
component
changes sign
velocity,
momentum,
mass flow
Tensor
invariant
Components
change
Four
components
change sign
Momentum
flow
Pseudovector
invariant
Components
change
Two
components
change sign
Curl of speed,
angular speed
Vector calculus - basics
A vector – standard notation
for three dimensions




A  ( Ax , Ay , Az )  Ax i  Ay j  Az k
Unit vectors i,j,k are vectors of magnitude 1 in directions of the x,y,z
axes.



i  (1,0,0)
Magnitude of a vector
j  (0,1,0) k  (0,0,1)

2
2
2
A  A  Ax  Ay  Az
Position vector is a vector r from
the origin to the current position




r  ( x, y , z )  x i  y j  z k
where x,y,z, are projections of r
to the coordinate axes.
Adding and subtracting vectors
 
A  B  ( Ax  Bx , Ay  By , Az  Bz ),
 
A  B  ( Ax  Bx , Ay  By , Az  Bz ),

A  ( Ax , Ay , Az )

B  ( Bx , B y , Bz )
Multiplying a vector by a scalar

k  A  (k Ax , k Ay , k Az ),
Example of multiplying of a vector by a scalar in a plane

u  (2,1)


v  2  u  2  (2,1)  (4,2)
Example of addition of three vectors in a plane
The vectors are given:



u  (2,1); v  (2,3); w  (2,0)
Numerical addition gives us
   
z  u  v  w  (2  2  (2),1  3  0)  (2,4)
Graphical solution:
Example of subtraction of two vectors a plane
The vectors are given:


u  (2,3); v  (1,2)
Numerical subtraction gives us
  
z  u  v  (2  (1),3  2)  (3,1)
Graphical solution:
Scalar product
Scalar product (dot product) – is defined as
Where Θ is a smaller angle between vectors
a and b and S is a resulting scalar.
   
a  b  a  b  cos   S
  n
a  b   ai bi  S
i 1
For three component vectors we can write
 
S  a  b  ax  bx  a y  by  az  bz  ab  cos 
Geometric interpretation – scalar product is
equal to the area of rectangle having a and
b.cosΘ as sides. Blue and red arrows
represent original vectors a and b.
Basic properties of the scalar product
   
a b  b  a
 
 
a  b  a b  0
   
a b a  b  ab
Vector product
Vector product (cross product) – is defined as
Where Θ is the smaller angle between vectors
a and b and n is unit vector perpendicular to the
plane containing a and b.
Geometric interpretation - the magnitude of the
cross product can be interpreted as the positive
area A of the parallelogram having a and b as
sides
 

a  b  ab  sin   n
 
A  a  b  ab  sin 
Component notation

i

j

k
  
c  a  b  ax
ay
az 
bx
by
bz



 (a y bz  a z by )i  (a z bx  a x bz ) j  (a x by  a y bx )k
Basic properties of
the vector product
 
 
a  b  b  a
 
 
a  b  a  b  ab
   
a b a b  0
Direction of the resulting vector of the vector product
can be determined either by the right hand rule or by the screw rule
Vector triple product
     
  
a  (b  c )  b (a  c )  c (a  b )
Scalar triple product
     
  
a  (b  c )  b  (c  a )  c  (a  b )
ax
  
a  (b  c )  bx
ay
az
by
bz  V
cx
cy
cz
Geometric interpretation of
the scalar triple product is a
volume of a paralellepiped V
Operators
in scalar and vector fields
• gradient of a scalar field
• divergence of a vector field
• curl of a vector field
GRADIENT OF A SCALAR FIELD
We will denote by f(M) a real function of a point M in an area A. If A is two
dimensional, then
f M   f  x, y 
and, if A is a 3-D area, then
f M   f  x, y, z 
We will call f(M) a scalar field defined in A.
Let f(M) = f(x,y). Consider an equation f(M) = c. The curve defined by this
equation is called a level line (contour line, height line) of the scalar field f(M).
For different values c1, c2, c3, ... we may get a set of level lines.
y
x
c3
c2
c1
c4
c5
Similarly, if A is a 3-D area, the equation f(M) = c defines a surface called a
level surface (contour surface). Again, for different values of c, we will
obtain a set of level surfaces.
z
c1
c2
c3
c4
x
y
Scalar fields – Magnitudes
•
•
•
•
•
•
•
•
•
•
Temperature
Pressure
Gravity anomaly
Resistivity
Elevation
Maximum wind speed (without directional info)
Energy
Potential
Density
Time…
Scalar field example: Temperature distribution in a
multitubular reactor
Scalar field example #2: Heat exchangers
Directional derivative
Let F(M) be a 3-D scalar field and let us construct a value that
characterizes the rate of change of F(M) at a point M in a direction given
by the vector
e = (cos , cos , cos ) (the direction cosines).
F M 1 
z
M1
F M 
M


x

y
F M 1   F M 
F
 lim
e M 1  M
MM 1
Let M = [x,y,z] and M1=[x+x, y+y, z+z ], then
F M 1   F M   F  x  x, y  y, z  z   F  x, y, z  
F
F
F
 dF  x, y, z    
x 
y 
z 
x
y
z
F
F
F

 cos 
 cos  
 cos   
x
y
z
where
  MM 1
and
 0
as
  0.
This gives us
F
F
F
 F

 lim cos  
cos  
cos    
e  0 x
y
z

However, the first three terms of the limit do not depend on  and
so that
  0 as   0
F F
F
F

cos 
cos  
cos 
e x
y
z
Clearly,
F
e
assumes its greatest value for
F
F
F
e
i
j
k
x
y
z
This vector is called the gradient of F denoted by
F
F
F
grad F  x, y, z  
i
j
k
x
y
z
or, using the Hamiltonian operator
grad F  F
 nabla  :
Thus grad F points in the direction of the steepest increase in F or the steepest
slope of F.
Geometrically, for a c, grad F at a point M is parallel the unit normal n at M of
the level surface
F  x, y, z   c
z
grad F
F  x, y, z   c
n
M
y
x
Scalar field and gradient
Scalar field associates a scalar quantity to every point in a space. This
association can be described by a scalar function f and can be also time
dependent. (for instance temperature, density or pressure distribution).


S (r , t )  f (r , t )
The gradient of a scalar field is a vector field that points in the direction of the
greatest rate of increase of the scalar field, and whose magnitude is that rate
of increase.
 S  S  S
grad S  S  i
j
k
x
y
z
Example: the gradient of the function
f(x,y) = −(cos2x + cos2y)2 depicted as
a projected vector field on the bottom
plane.
Example 2 – finding extremes of the scalar field
Find extremes of the function:
Extremes can be found by assuming:
h( x, y)  xe

grad (h)  0
( x 2  y 2 )
In this case :
h h 
grad (h)  ( , )  0
x y
h
( x 2  y 2 )
2 ( x 2  y 2 )
e
 2x e
0
x
h
( x 2  y 2 )
 2 xye
0
y
e
( x2  y 2 )
2 ( x2  y 2 )
 2x e
y0
1
1  2x  x  
2
2
Answer: there are two extremes
1
1
h1  (
,0); h2  (
,0)
2
 2
Vector operators
Gradient
(Nabla operator)
Divergence
Curl
 S  S  S
grad S  S  i
j
k
x
y
z

  Ax  Ay  Az
div A    A  i
j
k
x
y
z



i
j
k

 

   Az Ay 
 
curl A    A 
 i 

x y z
z 
 y
Ax Ay Az
  Ax Az
 j

x
 z
Laplacian
   Ay Ax 


  k 
y 
  x
2S 2S 2S
S   S  div grad S  2  2  2
x
y
z
2
Vector field, vector lines
Let a vector field f(M) be given in a 3-D area , that is, each
M  Ω is assigned the vector
f M   f1  x, y, z  i  f 2  x, y, z  j  f 3  x, y, z  k
A vector line l of the vector field f(M) is defined as a line with the
property that the tangent vector to l at any point L of l is equal to f(L).
Vector fields – Magnitude and direction
• Includes displacement, velocity, acceleration, force,
momentum…
• Magnetic field (Scale Earth or mineral)
• Water velocity field
• Wind direction on a weather map
• Electric field
Vector fields
VELOCITY
FIELDS
The speed at any given point is
indicated by the length of the arrow.
This means that if we denote by
dS  dx i  dy j  dz k
the tangent vector of l, then, at each point M, the following equations
hold
dx
dy
dz


f1  M  f 2  M  f 3  M 
This yields a system of two differential equations that can be used to
define the vector lines for f(M). This system has a unique solution if f1, f2,
f3 together with their first order partial derivatives are continuous and do
not vanish at the same point in . Then, through each
, there
passes exactly one vector line.
Ω
M Ω
Note: For a two-dimensional vector field we similarly get the differential
equation
dx
dy

f1  M  f 2  M 
Example
Find the vector line for the vector field
f M   x i  y j  2 z k
passing through the point M = [1, -1, 2].
Solution
We get the following system of differential equations
dx dy
dz


x  y  2z
dx
dy

x
y
or
dy dz

y 2z
this clearly has a solution
xy  c1 ,
y 2  c2 z
or, using parametric equations,
c1
t2
x  , y  t , z  , t  1
t
c2
Since the resulting vector line should pass through M, we get
1
c1  1, c2 
2
1
x   , y  t , z  2t 2 , t  1
t
Example
Find the vector lines of a planar flow of fluid characterized by the field of
velocities f(M)=xy i - 2x(x-1) j.
Solution
The differential equation defining the vector lines is
dx
dy

xy
2 x x  1
integrating this differential equation with
separated variables, we obtain
1 2
2  x  1 dx    y dy  x  2 x   y  c1
2
2
y
 x  12   c c  0
which yields
2
2
y
c=4
c=3
c=2
c=1
x
VECTOR FIELD
We will denote by f(M) a real vector function of a point M in an area A. If A
is two dimensional, then
f M   f1 x, y i  f 2 x, y  j
and, if A is a 3-D region, then
f M   f 1 x, y i  f 2 x, y  j  f 3 x, y k
We will call f(M) a vector field defined in A.
Flux through a surface
Let  be a simple (closed) surface and f(M) a 3-D vector field. The surface
integral
 f M  dS   f x, y, z  dydz  f x, y, z  dxdz  f x, y, z  dxdy
1
2
3
is referred to as the flux of the vector field f(M) through the surface
.
Divergence of a vector field
The arrows in the box represent velocity vectors
There is only inflow to the red box: Sink
There is only outflow from the blue box: Source
Divergence of a vector field
closed 3-D area V with S
as border
vector field f x, y, z 
S = V
M
V
D
 f x, y, z dS
an internal point M
|V| is the volume of V
V
V
D is the flux of f(x, y, z) through V per unit volume
Let us shrink V to M, that is, the area V becomes a point and see what D
does. If f (x, y, z) has continuous partial derivatives, the below limit exists, and
we can write symbolically:
DM   lim D  lim
V M
 f x, y, z dS
V
V M
V
If we view f (x, y, z) as the velocity of a fluid flow, D(M) represents the rate
of fluid flow from M.
for D(M) > 0, M is a source of fluid;
for D(M) < 0, M is a sink.
if D(M) = 0, then no fluid issues from M.
If we perform the above process for every M in the region in which the vector
field f (x ,y, z) is defined, we assign to the vector field f (x, y, z) a scalar field
D(x,y,z) = D(M).
This scalar field is called the divergence of f (x, y, z).
We use the following notation:
DM   div f x, y, z   div  f1 x, y, z i  f 2 x, y, z  j  f 3 x, y, z k 
It can be proved that, in Cartesian coordinates, we have
div f x,y,z 



f1 x, y, z  
f 2 x, y, z  
f 3 x, y, z 
x
y
z
Or, using the Hamiltonian or nabla operator



 i
j k
x
y
z
we can write
div f x, y, z     f x, y, z 
Gauss's - Ostrogradski's theorem
Let us take a closed surface  that contains a 3D region V where a vector field
f (x, y, z) is defined and "add-up" the divergence of f (x, y, z) within V, that is,
calculate the tripple integral
 div f x, y, z  dx dy dz
V
From what was said about the divergence being the rate of flow through
points of 3D space, we could conclude that this integral should represent
what flows through  as the boundary of V. However this is actually the flux
of f (x, y, z) through  or, in symbols
 f x, y, z dS
This is what Gauss's - Ostrogradsky's theorem says:
If a vector function f(x, y, z) has continuous partial derivatives in a 3D region V
bounded by a finite boundary , we can write
 f x, y, z  dS   div f x, y, z  dx dy dz
V
where the surface on the left-hand side of the equation is oriented so
that the normals point outwards.
The most general form of this formula was first proved by Mikhail
Vasilevich Ostrogradsky in 1828.
The curl of a vector field

n
A
M
l
vector field f (x, y, z)
For a plane  determined by its unit normal n, containing a closed curve l with a
fixed point M inside the area A bounded by l, define the quantity
C n , l , M  
 f x, y, z  ds
l
A
|A| is the surface area
of A
If f (x, y, z) has continuous partial derivatives, we can calculate
C n , M   lim C n , l , M 
l M
and C (n, M) does not depend on the choice of l and the way it
shrinks to M. It is only determined by the point M and the direction of
n
It can further be proved that there exists a universal vector c(M) such that
cM   n  C n , M 
for every normal n determining the plane 
Using the above method we can assign a vector c(M) to every point M in the
3D-region in which the vector field satisfies the assumptions (continuous partial
derivatives). In other words, we have defined to the original vector field f (x, y, z)
a new vector field c (x, y, z).
This vector field is called the curl of f (x, y, z) and denoted by
curl f (x, y, z) or rot f (x ,y, z)
If f (x, y, z) = f1(x, y, z) i + f2(x, y, z) j + f3(x, y, z) k, it can be shown that
 f 2 f 3   f 3 f1   f1 f 2 
i  
k
curl f x, y, z   



 j  
 z y   x z   y x 
The last formula can be written using the following formal determinant
i



curl f x, y, z 
x
f1 x, y, z 
j

y
f 2  x, y , z 
or
curl f x, y, z     f x, y, z 
k

z
f 3  x, y , z 
Suppose the vector f (x, y, z) field represents a flow of fluid.
Let us think of a very small turbine on a shaft positioned at a point M.
Through the fluid flow, the turbine will turn at a speed s.
Let us move the shaft changing its direction while leaving the turbine at the
point M.
In a certain direction c(M), the speed s of the turbine will reach its maximum.
Then c(M) is clearly the curl of f (x, y, z) at M.
s
M
s
s