{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

tensor_analysis_basics

tensor_analysis_basics - MY5400 MIDTERM REVIEW FALL 2011...

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

MY5400 MIDTERM REVIEW, FALL 2011, PATRICK K BOWEN In tensor notation, indices with only one occurrence are free indices , and those with two occurrences are dummy indices . More than two are not allowed. Free indices can assume any value from 1 3, and dummy indices must be summed over. The Kronecker Delta has a simple form: { The Kronecker Delta will switch the indices , in the term that comes after it ( will switch the existing index “i" with new index “j” that is: The Kronecker Delta can be factored “out” of any vector or matrix, say to turn . The Permutation Symbol , , will assume a value of 1, -1, or 0 depending on the index order and value. This index will assume values: { The even/odd permutation in determined by drawing a circle in which 1,2,3 are ordered clockwise . If the index is, say, 2,3,1, then the permutation will be even , because the indices follow the clockwise arrangement. If the index is 1,3,2, then the permutation will be odd because the indices run counter to the circle direction. If the index is 1,3,1, then the index will assume a value of zero. Remember that indices in different equations are arbitrary , that is: Recall that the directional cosine of a vector is the vector’s components divided by the vector’s length: ||⃑ || The dot product is the amount of vector in the direction of vector , or: ⃑ ⃑ ||⃑ ||||⃑ || ( ⃑⃑ ⃑⃑ ) The cross product is the vector ( ⃑⃑⃑ ) orthogonal to the plane formed by and in a right-handed manner: ⃑ ⃑ ( ⃑⃑ ⃑⃑ ) ⃑⃑⃑ The product of two permutation indices can be written instead in terms of Kronecker Deltas: In this identity, the dummy index must be written first. The two permutation indices can be rewritten as long as the index sense

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

tensor_analysis_basics - MY5400 MIDTERM REVIEW FALL 2011...

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

View Full Document
Ask a homework question - tutors are online