HW1 - x 2 axis. What is the magnitude of r A in each case?...

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AE 321 – Homework #1 Due: Monday September 8, 2008 Chapter 1: Mathematical Preliminaries 1. Write out the following using standard (longhand) notation A ij B jk = C ik A i = B i , k C k 2. Compute the following expressions (where ± is the Kronecker delta): ij ij = ? ij jk ki = ? 3. Given the identity, ijk pqr = det ² ip iq ir jp jq jr kp kq kr ³ ´ µ µ µ µ · ¸ ¸ ¸ ¸ where is the alternator and is the Kronecker delta, use indicial notation to find the simplest expressions for: (i) ijk iqr = (ii) ijk ijr = (iii) ijk ijk = . 4. If a , b and c are vectors, use indicial notation to prove the following vector identities: a ± b () c = a b ± c () a ± b () ± c = a c () b- b c () a 5. Let r A = 2 r e 1 + 4 r e 2 + 2 r e 3 . Find the components of r A after performing a counter clockwise rotation of 45° around the
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Unformatted text preview: x 2 axis. What is the magnitude of r A in each case? 6. The dyadic product between two vectors a and b is defined as: a b ( ) ij = a i b j . Using indicial notation prove the following (where W is a second order tensor): a b ( ) W = a W T b ( ) 7. Quantities a , u and W are a scalar, vector and two tensor field respectively (i.e. they are functions of position x 1 , x 2 , x 3 ). Prove the following identities using indicial notation: a = u ( ) = u ( ) 2 u W T u ( ) = u W ( ) + W u ( )...
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This note was uploaded on 03/07/2011 for the course AE 311 taught by Professor Dutton,j during the Spring '08 term at University of Illinois, Urbana Champaign.

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