Contra vs co vector 2013
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Contra vs co vector 2013 Contra vs co vector 2013 Document Transcript

  • 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
  • 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.
  • 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.
  • 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:
  • 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
  • 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