2. 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 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" 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 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.

4. 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

5. Temperature: linear scale
But pH scale is logarithmic!

6. 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.

7. 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.

9. Frame of reference example

11. Frame of Reference
Transformations
Translation
Rotation
Reflection

12. 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.

13. Properties of scalar, vector and tensor quantities for coordinate transformations
Physical quantity
Translation
Rotation
Reflection
Example
Scalar
invariant
Mass , temperature
Vector
Components change
One component changes sign
velocity, momentum, mass flow
Tensor
Four components change sign
Momentum flow
Pseudovector
Two components change sign
Curl of speed, angular speed

14. Vector calculus - basics
A vector – standard notation for three dimensions
Unit vectors i,j,k are vectors of magnitude 1 in directions of the x,y,z axes.
Magnitude of a vector
Position vector is a vector r from the origin to the current position
where x,y,z, are projections of r to the coordinate axes.

15. Example of multiplying of a vector by a scalar in a plane
Adding and subtracting vectors
Multiplying a vector by a scalar
Example of multiplying of a vector by a scalar in a plane

16. Example of addition of three vectors in a plane
The vectors are given:
Numerical addition gives us
Graphical solution:

17. Example of subtraction of two vectors a plane
The vectors are given:
Numerical subtraction gives us
Graphical solution:

18. 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.
For three component vectors we can write
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

19. 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
Component notation
Basic properties of the vector product

20. 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
Geometric interpretation of the scalar triple product is a volume of a paralellepiped V
Scalar triple product

21. Operators in scalar and vector fields
gradient of a scalar field
divergence of a vector field
curl of a vector field

22. 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
and, if A is a 3-D area, then
We will call f(M) a scalar field defined in A.

23. Let f(M) = f(x,y). Consider an equation f(M) = c
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 c5 c4 c3 c2 c1

24. 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 y c4 x

25. Scalar fields – Magnitudes
Temperature
Pressure
Gravity anomaly
Resistivity
Elevation
Maximum wind speed (without directional info)
Energy
Potential
Density
Time…

26. Scalar field example: Temperature distribution in a multitubular reactor

27. Scalar field example #2: Heat exchangers

28. 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). z M1 M y x

29. Let M = [x,y,z] and M1=[x+x, y+y, z+z ], thenas
where

30. This gives us
However, the first three terms of the limit do not depend on  and so that

31. Clearly,
assumes its greatest value for
This vector is called the gradient of F denoted by
or, using the Hamiltonian operator

32. 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 x z y
grad F n M

33. 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).
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.
Example: the gradient of the function f(x,y) = −(cos2x + cos2y)2 depicted as a projected vector field on the bottom plane.

34. Example 2 – finding extremes of the scalar field
Find extremes of the function:
Extremes can be found by assuming:
In this case :
Answer: there are two extremes

35. Vector operators
Gradient
(Nabla operator)
Divergence
Curl
Laplacian

36. Vector field, vector lines
Let a vector field f(M) be given in a 3-D area , that is, each
is assigned the vector
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).

37. 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

38. Vector fields VELOCITY FIELDS
The speed at any given point is indicated by the length of the arrow.

39. This means that if we denote by
the tangent vector of l, then, at each point M, the following equations hold
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.
Note: For a two-dimensional vector field we similarly get the differential equation

40. Example
Find the vector line for the vector field
passing through the point M = [1, -1, 2].
Solution
We get the following system of differential equations or
this clearly has a solution
or, using parametric equations,

41. Since the resulting vector line should pass through M, we get

42. 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
integrating this differential equation with separated variables, we obtain
which yields

43. y
c=4
c=3
c=2
c=1 x

44. 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
and, if A is a 3-D region, then
We will call f(M) a vector field defined in A.

45. Flux through a surface
Let  be a simple (closed) surface and f(M) a 3-D vector field. The surface integral
is referred to as the flux of the vector field f(M) through the surface .

46. 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

47. Divergence of a vector field
closed 3-D area V with S as border
S = V M
an internal point M V
|V| is the volume of V
D is the flux of f(x, y, z) through V per unit volume

48. 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:
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.

49. 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:

50. It can be proved that, in Cartesian coordinates, we have
Or, using the Hamiltonian or nabla operator
we can write

51. 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
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

52. 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
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.

53. 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
|A| is the surface area of A

54. If f (x, y, z) has continuous partial derivatives, we can calculate
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
for every normal n determining the plane 

55. 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

56. The last formula can be written using the following formal determinant

57. 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 s M s