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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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2

(We change the dummy indices )

: (2 )and (1 )Subtracting relations :

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4 If we now :

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 :