Contra vs co vector 2013

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

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Contra vs co vector 2013

  1. 1. Contra Variant and CoVariant Tensor and VectorDifference between them2013Umaima_AyanSession 2009-13Submitted By: Atiqa Ijaz KhanRoll no: ss09-03Subject: Riemannian geometrySubmitted To: Sir JunaidDated: 28th– May-2013
  2. 2. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]1 | Session 2009-13Table of Contents1. Introduction to the Tensor 022. Contra variant Vector 023. Co variant Vector 034. Differences between both types 045. References 06
  3. 3. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]2 | Session 2009-13Introduction to the TensorsTensors are defined by means of their properties of transformation under thecoordinate transformation.Vectors are the special case of the tensors.Contra variant TensorsConsider two neighboring points P and Q in the manifold whose coordinates arexr and xr + dxr respectively. The vector PQ is then described by the quantitiesdxr which are the components of the vector in this coordinate system. In thedashed coordinates, the vector PQ is described by the components d xrwhichare 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 componentsof a contra variant vector if they transform, on change of coordinates, accordingto the equation:T r  x rxs Ts.Where the partial derivatives are evaluated at the point P.
  4. 4. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]3 | Session 2009-13Definition:A set of n 2 quantities T rs associated with a point P are said to be thecomponents of a contra variant tensor of the second order if they transform, onchange of coordinates, according to the equation:T rs  x rxm x sxn T mn.Obviously the definition can be extended to tensors of higher order. A contravariant 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 tensorsLet φ be an invariant function of the coordinates, i.e. its value may depend onposition P in the manifold but is independent of the coordinate system used.Then the partial derivatives of φ transform according to: xr xsxs xrThe partial derivatives of an invariant function provide an example of thecomponents of a covariant vector.
  5. 5. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]4 | Session 2009-13Definition:A set of n quantities Tr associated with a point P are said to be the componentsof a covariant vector if they transform, on change of coordinates, according tothe equation:Tr xs xr Ts.Extending the definition as before, a covariant tensor of the second order isdefined by the transformation:Trs xm xrxn xs TmnAnd similarly for higher orders.Differences between these TypesThe few of the differences between contra variant and co variant tensors are asfollows:SerialNo.Contra variant Tensor Co variant Tensor01. Writing the components with theSubscriptWriting the components with theSuperscript02. The tensor is represented by thecomponents in the “direction ofcoordinate increases”The tensor is represented by thecomponents in the “directionorthogonal to constant coordinatesurfaces”03. Examples: Examples:
  6. 6. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]5 | Session 2009-131. Velocity2. Acceleration3. Differential Position d=ds1. Gradient of scalar field
  7. 7. May 28, 2013 [CONTRA VARIANT AND CO VARIANT TENSOR AND VECTOR]6 | Session 2009-13References1. Matrices And Tensors In PhysicsBy A W Joshi2. Introduction to Tensor Calculus, Relativity, and CosmologyBy D. F. Lawden

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