differential geometry w notes from teacher_Part_23

differential geometry w notes from teacher_Part_23 - E * |...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
2.4. TENSORS 45 The dimension of the vector space T p is dim T p = n p . The tensor product of two covectors α, β E * is a covariant tensor α β E * E * of rank 2 defined by: v , w E ( α β )( v , w ) = α ( v ) β ( w ) . The components of the tensor product α β are ( α β ) i j = α i β j . The tensor product of a covariant tensor Q of rank p and a covariant ten- sor T of rank q is a covariant tensor Q T of rank ( p + q ) defined by: v 1 , . . . , v p , w 1 , . . . , w q E ( Q T )( v 1 , . . . , v p , w 1 , . . . , w q ) = Q ( v 1 , . . . , v p ) T ( w 1 , . . . , w q ) . The components of the tensor product Q T are ( Q T ) i 1 ... i p j 1 ... j q = Q i 1 ... i p T j 1 ... j q . Thus, : T p × T q T p + q . Tensor product is associative . The basis in the space T p is σ i 1 ⊗ ··· ⊗ σ i p , where 1 i 1 , . . . , i p n . A covariant tensor Q of rank p has the form Q = n X i 1 ,..., i p = 1 Q i 1 ... i p σ i 1 ⊗ ··· ⊗ σ i p . di geom.tex; April 12, 2006; 17:59; p. 48
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
46 CHAPTER 2. TENSORS 2.4.2 Contravariant Tensors A contravariant vector can be considered as a linear real-valued functional v : E * R . A contravariant tensor of rank p (or a tensor of type ( p , 0)) is a multi- linear real-valued functional T : E * × ··· ×
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: E * | {z } p R Remarks. The function T ( 1 , , p ) is linear in each argument. The functional T is independent of any basis. A contravariant vector (covector) is a contravariant tensor of rank 1. The components of the tensor T with respect to the basis i are dened by T i 1 ... i p = T ( i 1 , . . . , i p ) . Then for any covectors ( a ) = n X j = 1 ( a ) j j , where a = 1 , . . . , p , we have T ( (1) , . . . , ( p ) ) = n X j 1 ,..., j p = 1 T j 1 ... j p (1) j 1 ( p ) j p . The inverse matrix of the components of a metric tensor denes a con-travariant tensor g-1 of rank 2 by g-1 ( , ) = n X i , j = 1 g i j i j . di geom.tex; April 12, 2006; 17:59; p. 49...
View Full Document

Page1 / 2

differential geometry w notes from teacher_Part_23 - E * |...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online