Tensors are mathematical objects that transform
in a special way when the basis vectors change.
Understanding tensors is fundamental to understanding
the curvature of space time in Einstein’s General Relativity.
A tensor of rank 1 is a vector.
Let’s pick new basis vectors.
We typically describe a vector by how many of each of
the basis vectors we have to add together to produce it.
But, there is also another way to describe a vector in terms of the basis vectors.
This is by taking the dot product of the vector with each of the basis vectors.
For now, let us return to the typical way of describing a vector.
Here, the vector has the components 4, 2, and 6.
Suppose we double the length of each of the basis vectors.
The components of this vector are now 2, 1, and 3,
which means that the components have decreased.
When we increased the lengths of the basis vectors,
the components of our vector decreased.
If we decrease the lengths of the basis vectors,
the components of our vector increase.
Because these two quantities change “contrary” to one another, we refer to this
by saying that these are the “contra-variant” components of the vector.
Describing a vector in terms of the “contra-variant” components
is the way we usually describe a vector.
Suppose we instead describe the vector in terms of
its dot product with each of the basis vectors.
If the basis vectors are of length 1, then the result of
these dot products will be these three lengths.
If we double the length of a basis vector,
then its associated dot product will also double.
When we increase the lengths of the basis vectors, the dot products also increase.
When we decrease the lengths of the basis vectors, the dot products also decrease.
Because both quantities vary in the same way, we refer to this by saying that
these are the “co-variant” components for describing the vector.
We can distinguish the variables for the “co-variant” components
from variables for the “contra-variant” components by using
sub-scripts instead of super-scripts for the index values.
We can distinguish the variables for the “co-variant” components
from variables for the “contra-variant” components by using
sub-scripts instead of super-scripts for the index values.
Since these are just two different ways to describe the same vector,
the same variable name is used, which in this case is the variable “V”.
Suppose we have two different vectors.
Let’s call the first vector “V” and call the second vector “P”,
and let’s always continue to use these names, even if the basis vectors change.
Suppose we multiply one of the contra-variant components of V
with one of the contra-variant components of P.
If we consider every possible way in which we can do this,
we would get a matrix as shown.
This is an example of a tensor of rank 2 with two “contra-variant” index values.
Suppose we multiply the “co-variant” components of V
with “contra-variant” components of P as shown.
This will give us a different description of the same rank 2 tensor,
with one “co-variant” index value and one “contra-variant” index value.
Suppose we now instead multiply two co-variant components
from each of the two vectors as shown.
This will give us a different description of the same rank 2 tensor,
with two “co-variant” index values.
What makes a tensor a tensor is that when the basis vectors change, the components
of the tensor would change in the same manner as they would in one of these objects.
A tensor does not necessarily have to be created
from vector components as is shown in these examples.
A tensor of rank 1 is a vector.
A tensor of rank 1 has a number associated with each of the basis vectors.
In a tensor of rank 2, instead of associating a number with each basis vector,
we associate a number with every possible combination of two basis vectors.
In a tensor of rank 3, we associate a number
with every possible combination of three basis vectors.
A tensor of rank 3 can be composed of the combinations of components of three vectors.
We can create different descriptions of this tensor by using different combinations
of contra-variant and co-variant index values for the components of the three vectors.