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s11f1013

# s11f1013 - Solution for assignment 13(dr)2 = = = 2(v = =...

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Solution for assignment 13: ( d~ r ) 2 = ( dx ) 2 + ( dy ) 2 + ( dz ) 2 (1) = ( ) 2 + ρ 2 ( ) 2 + ( dz ) 2 (2) = ( dr ) 2 + r 2 sin 2 θ ( ) 2 + r 2 ( ) 2 (3) ( ~v ) 2 = ( ˙ x ) 2 + ( ˙ y ) 2 + ( ˙ z ) 2 (4) = ( ˙ ρ ) 2 + ρ 2 ( ˙ φ ) 2 + ( ˙ z ) 2 (5) = ( ˙ r ) 2 + r 2 sin 2 θ ( ˙ φ ) 2 + r 2 ( ˙ θ ) 2 (6) Locally orthonormal unit vectors are given by ˆ ρ = cos( φ ) ˆ x + sin( φ ) ˆ y (7) ˆ φ = - sin( φ ) ˆ x + cos( φ ) ˆ y (8) ˆ r = sin( θ ) ˆ ρ + cos( θ ) ˆ z = sin( θ ) cos( φ ) ˆ x + sin( θ ) sin( φ ) ˆ y + cos( θ ) ˆ z (9) ˆ θ = cos( θ ) ˆ ρ - sin( θ ) ˆ z = cos( θ ) cos( φ ) ˆ x + cos( θ ) sin( φ ) ˆ y - sin( θ ) ˆ z (10) Usefull is table 1 of dot products between the Cartesian and the new, local orthonor- mal unit vectors. ˆ x ˆ y ˆ z ˆ ρ cos φ sin φ 0 ˆ φ - sin φ cos φ 0 ˆ r sin θ cos φ sin θ sin φ cos θ ˆ θ cos θ cos φ cos θ sin φ - sin θ Table 1: Dot products between Cartesian and local orthonormal unit vectors. The velocity vector is in Cartesian, cylindrical and spherical coordinates

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s11f1013 - Solution for assignment 13(dr)2 = = = 2(v = =...

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