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HW1 - x 2 axis What is the magnitude of r A in each case 6...

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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 ² iq ² ir ² jp ² jq ² jr ² kp ² kq ² kr ³ ´ µ µ µ µ · ¸ ¸ ¸ ¸ where ² 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
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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: ± ² ± a = ± ² ± ² u ( ) = ± ± • u ( ) ³ ± 2 u ± • W T u ( ) = u • ± • W ( ) + W • ± u ( )...
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