{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

IndexNotationHints

# IndexNotationHints - b j e j = a i b j e i × e j = a i b j...

This preview shows page 1. Sign up to view the full content.

ETH Zurich Department of Mechanical and Process Engineering Institute for Mechanical Systems Center of Mechanics Prof. Dr. Edoardo Mazza 1 Useful rules for index notation The permutation ε ijk is defined as follows, ε ijk = 1 (i,j,k)=(123),(231),(312) - 1 (i,j,k)=(132),(213),(321) 0 (all other combinations) . (1) Such that when the indices are in increasing order the value is postive 1, and when the indices are in decreasing order the value is negative 1. Otherwise it is 0. This definition is motivated by the need for defining an easy way to write the cross product. e i × e j = ε ijk e k . (2) Given this notation, we can easily express the cross product between vectors a and b as, a × b = ( a i e i ) × ( b j
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( b j e j ) = a i b j e i × e j = a i b j ε ijk e k (3) Remarks: The Kronecker delta can be used as identity matrix: I ˆ= δ ij (4) The permutation has the properties that, ε ijk = ε jki = ε kij (5) and, ε ijk =-ε jik . (6) There is exists the following relation between the permutation and the Kronecker delta, 1. ε ijk ε pqk = δ ip δ jq-δ iq δ jp (7) Here i,j,p,q are free indices and must match on both sides of the equation. k is a dummy index. 2. ε ijk ε pjk = 2 δ ip (8) Here i,p are free indices and must match on both sides of the equation. j,k are dummy indices. 3. ε ijk ε ijk = 6 (9) 1...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern