Contra-varaint tensor differences as compare to Co-vararint tensor

1. 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

2. 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

3. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]
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 PQ is then described by the quantities
dxr which are the components of the vector in this coordinate system. In the
dashed coordinates, the vector PQ is described by the components d x
r
which
are related to dxr by equation as follows:
d x r 
 x r
x
m 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
x
s Ts
.
Where the partial derivatives are evaluated at the point P.

4. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]
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
x
m
 x s
x
n T mn
.
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 

x
s
xs
 x
r
The partial derivatives of an invariant function provide an example of the
components of a covariant vector.

5. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND 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:
Tr 
xs
 x
r Ts
.
Extending the definition as before, a covariant tensor of the second order is
defined by the transformation:
Trs 
xm
 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:

6. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]
5 | Session 2009-13
1. Velocity
2. Acceleration
3. Differential Position d=ds
1. Gradient of scalar field

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