What is this "Tensor" any way
?
In Physics you find Tensors almost everywhere
. In Newtonian mechanics, in physics of materials, in electromagnetism,
in relativity and in many other applications that i can't think of or i
don't know . So you can understand the importance of Tensors in Physics
. But the question remains, what is a Tensor ? One can very easily get
confused by the indices that some times appear at the upper right of the
component and others at the lower right as subscripts or the free indices
and the dummy indices and all the other conventions that are made in most
of the books that look very confusing if you are not used to them . As
a result one is very likely to miss the point and at the end not understand
what a Tensor is . In reality things are much more simple, a
Tensor is just a function that is linear and has values over |R .
But what does that mean? Let's consider the contravariant tensor A
with components Ai . This tensor corresponds to the vectors
we all know . We define the covariant tensor B
with components Bj as the function B=B(A)
which is linear and has values over |R, meaning that if we have the
contravariant tensors A and C then
we have :
B(aA+cC)=aB(A)+cB(C)=d
Î |R
This means that B(A)Î
|R and we define the way it acts on the contravariant tensor as :
B(A)=BiAi
=åBiAi =B1A1+B2A2
+B3A3 + ...
We can see that the action of B
on A doesn't depend on any other factor so we conclude
that it is invariant . We can see that for every contravariant tensor we
can define a space of covariant tensors (functions) which will have the
contravariant tensor as argument and will give value on |R . This
space is the dual space of the tensor . The dual space of a vector of a
vector-space generally is the space of functions from the vector
to |R . The concept of the dual space has many applications one of
which is at quantum mechanics . Now we can see that the contravariant tensor
A can be considered as a function which acts on the covariant
tensor B and gives values on |R . It is
necessary to define the base vectors of our space of covariant and contravariant
tensors . As you will see from the previous formulation and the condition
which the base vectors must satisfy (that is : ei(B)=Bi
)
the covariant base are the contravariant vectors eiand
the contravariant base are the covariant vectors
qj.
Now we can define the tensor product between two tensors as :
BÄC=F
The F tensor is not
the same tensor with the other two so i will use the symbol Fij
for it . Its components will be Fij=BiCj.
Generally we have that BÄC
is not equal to CÄB
. How is this tensor defined according to the previous formulation? It
is simple, the tensor of covariant rank two is the function with two contravariant
tensors of rank one as arguments which gives its values on |R .So
it is of the form :
Fij
=Fij (A,B)=åå
FijAiBj =FijAiBj
This way we can define all the tensors of any
rank covariant or contravariant and even mixed . This way some of the properties
of tensors become more clear, for example the property of the metric to
raise and lower indices :
gijA=gij(A,_)=å
gijAi =gijAi =Aj=A=A(_)
which is now a covariant tensor waiting for a
contravariant argument to take it to |R .
This presentation is a very simplified one but
i only wanted to make things a little more clear to you, i hope i have
helped a little .If not i am sorry. |
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