SF State PHYS 385 - Chapter 2. The Special Symbols, Einstein Summation Convention and Group Theory (12 pages)

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Chapter 2. The Special Symbols, Einstein Summation Convention and Group Theory



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Chapter 2. The Special Symbols, Einstein Summation Convention and Group Theory

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Lecture Notes


Pages:
12
School:
San Francisco State University
Course:
Phys 385 - Introduction To Theoretical Physics I

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Vector Spaces in Physics 8 16 2014 Chapter 2 The Special Symbols ij and ijk the Einstein Summation Convention and some Group Theory Working with vector components and other numbered objects can be made easier and more fun through the use of some special symbols and techniques We will discuss two symbols with indices the Kronecker delta symbol and the Levi Civita totally antisymmetric tensor We will also introduce the use of the Einstein summation convention References Scalars vectors the Kronecker delta and the Levi Civita symbol and the Einstein summation convention are discussed by Lea 2004 pp 5 17 Or search the web One nice discussion of the Einstein convention can be found at http www2 ph ed ac uk mevans mp2h VTF lecture05 pdf You may find other of the lectures at this site helpful too A The Kronecker delta symbol ij This symbol has two indices and is defined as follows 0 i j 2 1 ij Kronecker delta symbol i j 1 2 3 1 i j Here the indices i and j take on the values 1 2 and 3 appropriate to a space of threecomponent vectors A similar definition could in fact be used in a space of any dimensionality We will now introduce new notation for vector components numbering them rather than naming them This emphasizes the equivalence of the three dimensions We will write vector components as Ax 2 2 Ay Ai i 1 3 A z We also write the unit vectors along the three axes as 2 3 i j k e i i 1 3 The definition of vector components in terms of the unit direction vectors is Ai A e i i 1 3 The condition that the unit vectors be orthonormal is e i e j ij 2 4 2 5 This one equation is equivalent to nine separate equations i i 1 j j 1 k k 1 i j 0 j i 0 i k 0 k i 0 j k 0 k j 0 We have now stopped writing i j 1 3 it will be understood from now on that in a 3 dimensional space the free indices like i and j above can take on any value from 1 to 3 2 1 Vector Spaces in Physics 8 16 2014 Example Find the value of j k obtained by using equation 2 5 Solution We substitute e 2 for j and e 3 for k



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