On Friday, December 19, 2014 8:20:58 AM UTC+2, Alok Gahlot wrote:
>
> my question is what is the difference in contravariant and covariant
>> tensor
>>
> The simple (and stupid) answer is that they are objects of different type.
In more detail, their components behave differently under general coordinate
transformations.
> and how can we decide to choose tensor in any problem?
>
There is no freedom of choice in general. Only if an inner product
("metric") is given,
contravariant, covariant, and mixed tensors can be transformed into each
other.
Even then there is usually one form that is best suited to the problem in
question.
> what is the rule by which we know that we use covariant or contravariant
> or mixed tensor ?
>
This is a hard question. I cannot write down any general rule valid in all
situations.
Instead I try to give some examples in the case of tensors on a (single)
vector space V
as in http://en.wikipedia.org/wiki/Tensor_%28intrinsic_definition%29 .
Loosely speaking, a tensor can be regarded as a (linear or multilinear)
mapping,
often in more than one way.
A tensor is of type (p,q) if there are q vector arguments and p vector
values.
If the value is a scalar, then p = 0, and q = 0, if there are no
arguments.
1. A vector x in V is a tensor of type (1,0). It may be thought of as a
mapping with no arguments
and the single vector x as its value.
2. A covector x* in V* is a tensor of type (0,1). It is a linear mapping
from V to the scalars.
So its argument is a single vector and its value is a scalar.
3. A linear mapping u: V -> V is a tensor of type (1,1). It takes a single
vector x as its
argument and has another vector u(x) as its value.
4. An inner product (or metric) in V is a tensor of type (0,2). It has two
vector arguments x and y
and their scalar product x.y as the value.
5. A bilinear mapping VxV -> V is a tensor of type (1,2). It takes two
vector arguments
and the value is a vector. An example is the cross product in the
three-dimensional space.
Finally, if a metric is given, each vector x in V corresponds to a covector
x* in V*, namely the
linear form x*: y -> x.y . Then the difference between contravariance and
covariance disappears
under transformations that preserve the metric.
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