contra vs co vector 2013

Contra Variant and Co

Variant Tensor and Vector Difference between them

2013

Umaima_Ayan

Session 2009-13

Submitted By: Atiqa Ijaz Khan

Roll no: ss09-03

Subject: Riemannian geometry

Submitted To: Sir Junaid

Dated: 28th – May-2013

May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]

1 | Session 2009-13

Table of Contents

1. Introduction to the Tensor 02

2. Contra variant Vector 02

3. Co variant Vector 03

4. Differences between both types 04

5. References 06


2 | Session 2009-13

Introduction to the Tensors

Tensors are defined by means of their properties of transformation under the

coordinate transformation.

Vectors are the special case of the tensors.

Contra variant Tensors

Consider two neighboring points P and Q in the manifold whose coordinates are

xr and xr + dxr respectively. The vector P Q is then described by the quantities

dxr which are the components of the vector in this coordinate system. In the

dashed coordinates, the vector P Q is described by the components d x r

which

are related to dxr by equation as follows:

d x r x r

xm

dxm

.

The differential coefficients being evaluated at P.

Definition:

A set of n quantities T r associated with a point P are said to be the components

of a contra variant vector if they transform, on change of coordinates, according

to the equation:

T r x r

xs

Ts

.

Where the partial derivatives are evaluated at the point P.


3 | Session 2009-13

Definition:

A set of n 2 quantities T rs associated with a point P are said to be the

components of a contra variant tensor of the second order if they transform, on

change of coordinates, according to the equation:

T rs x r

xm

x s

xn

Tmn

.

Obviously the definition can be extended to tensors of higher order. A contra

variant vector is the same as a contra variant tensor of first order.

Definition:

A contra variant tensor of zero order transforms, on change of coordinates,

according to the equation:

T T ,

It is an invariant whose value is independent of the coordinate system used.

Covariant vectors and tensors

Let φ be an invariant function of the coordinates, i.e. its value may depend on

position P in the manifold but is independent of the coordinate system used.

Then the partial derivatives of φ transform according to:

x r

xs

xs

x r

The partial derivatives of an invariant function provide an example of the

components of a covariant vector.


4 | Session 2009-13

Definition:

A set of n quantities Tr associated with a point P are said to be the components

of a covariant vector if they transform, on change of coordinates, according to

the equation:

T rxs

x r

Ts.

Extending the definition as before, a covariant tensor of the second order is

defined by the transformation:

T rsxm

x r

xn

x s

Tmn

And similarly for higher orders.

Differences between these Types

The few of the differences between contra variant and co variant tensors are as

follows:

Serial

No.

Contra variant Tensor Co variant Tensor

01. Writing the components with the

Subscript

Writing the components with the

Superscript

02. The tensor is represented by the

components in the “direction of

coordinate increases”

The tensor is represented by the

components in the “direction

orthogonal to constant coordinate

surfaces”

03. Examples: Examples:


5 | Session 2009-13

1. Velocity

2. Acceleration

3. Differential Position d=ds

1. Gradient of scalar field


6 | Session 2009-13

References

1. Matrices And Tensors In Physics

By A W Joshi

2. Introduction to Tensor Calculus, Relativity, and Cosmology

By D. F. Lawden

contra vs co vector 2013

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