In a physical system at every point in space (x,y,z) we can observe physical quantities acting.
Some of these can be represented by simple numbers.
There are two types, we call these both scalars.
We can understand the two types best by considering a specific region of space.
Within the region
Extensive properties are additive. We add the numerical value at each point in the region to get one number that represents the property in that region.
For example the mass of our region is the sum of the mass values at every point.
Intensive properties
Are an average. We average the numerical value at every point to get a value to represent the property for the region.
For example temperature. We average, not add the temperatures at every point to get the region temperature.
We can write down the property as a function of position f(x,y,z)
This is known as a scalar field.
Some physical quantities are more complicated.
The simplest is where we assign or observe a vector at every point. Examples are velocity vectors in fluid flow, field vectors in electric and magnetic fields etc.
In our 3D coordinate system any vector can be written
[tex]v = \alpha x + \beta y + \gamma z[/tex]
Where the greek letters are numbers and roman letters are vectors.
So at every point there is a vector v which can be written in the above way. Each of the
[tex]\alpha x[/tex]; [tex]\beta y[/tex]; [tex]\gamma z[/tex]
are vectors in their own right and v is a linear combination of them. They are the basis vectors.
We call these components.
The important fact about components is that they are (linearly) independent. We can vary anyone without affecting one of the others.
We can write down a vector valued function v = f(x,y,z)
This is called a vector field.
But there are yet more complicated physical quantities, called Tensors, that we can assign or observe at any point in space.
The parts of tensors affect each other, unlike vector components, so we call these parts elements rahter than components.
Examples are the stress and strain tensors, the inertia tensor.
Again we can form a Tensor valued function whcih describes the distribution of the tensors in our 3D space.
T = f(x,y,z)
Since tensors are represented by 2 dimensional matrices the information of the interaction between the elements. However the elements may not form a basis - and don't in the case of the stress tensor.
You cannot in general change the stress on one axis, without affecting the stresses on the others.
this is a brief intorduction to physical quantities, without all the mathematical notation which tends to obscure the physical meaning.
Another way to look at it is to note that with vectors, there is no essential difference between its components. We could interchange x, y and z without altering things.
With tensors, on the other hand, we cannot do this. the elements are not all the same. We cannot interchange shear and normal stresses at will.
