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Covariant and Contravariant Vectors
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Transcript of Covariant and Contravariant Vectors
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Mathematical methods in physics to Prof . Khaled
abdelwaged
Of student / Hanan hassan makallawi
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Covariant And contra variant Vectors
A covariant vector is specifically a vector which transforms with the basis vectors, a
contravariant vector on the other hand is a vector that transforms against the basis vectors .
Contents
1-Introduction 2-What is the contra variant And covariant 3-From Vectors To Tensors
4- Algebraic properties of Tensors : 4-1 Collecting 4-1 multiplication
4-3 contraction
4-4 symmetric :
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1-Introduction
In multi linear algebra and tensor analysis, covariance and contra variance describe how the
quantitative description of certain geometric or physical entities changes with a change of
basis. For holonomic bases, this is determined by a change from one coordinate system to
another. When an orthogonal basis is rotated into another orthogonal basis, the distinction
between co- and contravariance is invisible. However, when considering more general
coordinate systems such as skew coordinates, curvilinear coordinates, and coordinate
systems on differentiable manifolds .
2-What is the contra variant And covariant
We Found that there are other vehicles for a vector
is called contra variant
is called covariant
These Compounds Transformed under the influence of coordinate
transformation To.
.
contra variant vector
.
covariant vector
Let us now find The dot product for them in the Cartesian coordinates
system
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dot product We see that the right border is not construed to
Now suppose that the vector U Turns into :
( contra variant vector )
And that the vector V Turns into :
( covariant vector )
dot product For them : If we take the
We find that happen, and this is due to the conversion of coordinates General
Coordinate Trans
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How do we know the length of the contra variant vector which has only upper
index ?
And also along the covariant vector which has only Lower index ?
3-From Vectors To Tensors
We have studied one type of vectors and is (dot product ) Let us now examine the
model gives more then on index like cross product
Example if we take the vector like covariant then the cross
product for them is and in the other coordinates system these
compounds have the form
and Thus
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: Only when there is in assembly
(upper index) with (lower index) as
in the case of (contra variant), the
output of the dot product does not
depend on the coordinate system
used .
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(We change the dummy indices )
: (2 )and (1 )Subtracting relations :
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Which form compounds vector The equation :
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To And gives the conversion from
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The cross product is the special case from tensor the and carrying
multiple indices some of these indices Be upper and some Be lower .
Usually, the tensor carrying multiple indices Like ( lower upper
lower + upper ) .
Of second rank : rtensoWe can write the
(contra variant)
(covariant)
(mixed )
Covariant Contravariant
Let's now symbolized types tensor (( Show that indicates the total rank tensor )) :
Scalar ( 0 , 0 )
Contra variant vector ( 1 , 0 )
Covariant tensor j rank 2 ( 0 , 2 )
Mixed tensor j rank 2 ( 1 , 1 )
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: 1Example
? -tensor What kind of
As we saw earlier , we wrote -tensor in the form of from type (0 , 2) if so , the converted is :
Means that dose not represent any of the transfers also , we can prove that do not represent the kind of ( 2 , 0) , but if we know it's kind of (1 , 1) :
It is the same transfers and that corresponds to tensor from kind mixed
4- Algebraic properties of Tensors :
contra variant covariant : Collecting 1-4
, S term of tensors from type (r ,s ) Then the total u = T + S T If the is :
(1)
And is tensor from type ( r ,s ) .
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Multiplied by the number of real ( ) in tensor givestensor of the same type :
(2)
Relationship ( 1 ) and ( 2 ) make tensors from the type(r ,s ) a vector
space
multiplication : 2-4
term of tensors from type ( , ) , S term of tensors T If the from type ( , ) and the holds beaten u = T x S
] ( + ) , ( + ) This is tensor from type [
Example :
Find multiplying tensor from type (2 , 1) and tensor from type (0 , 2) find conversion including product ?
Tensor from type (2 , 1) like
Tensor from type (2 , 1) like
And holds beaten before conversion is :
And holds beaten after conversion is
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Output it tensor from type (2 , 3)
:contraction 3 -4
If you give tensor from type (r ,s ) and we :
covariant index = contra variant index , and we collected all indices this process called
contraction and the output of the process is tensor from type ( r-1 , s-1 )
:Example
If you take tensor from type (2 , 1) and its compounds and made
(k=1) How is the final conversion of
?
This shows that turn out like compounds contra variant from type (1 , 0).
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:Example
From type (1 , 1) ? tensor What is the contraction of
Suppose the A is contra variant vector gives and B is covariant vector and gives
and the total multiply is :
This is Tensor from type (1 , 1) and when contraction we get :
This is holds multiply dot product of two vector and the output is scalar from
type (0 , 0)
4-4 symmetric : It is an important characteristic in physics and occur if others 2-indices then the
tensor output does not change or changes mark ( - ) , if not changed tensor with
change indices and it called symmetric , if change the negative sign it called
Antisummetric .
:Example If the T is Tensor from type ( 2 , 0 ) , U is Tensor of type ( 0 , 2 ) and it terms
symmetric
Antisymmetric
We can write any Tensor with asymmetric , Antisymmetric like ..
symmetric
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Antisymmetric
Collects another equations , we get :