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worksheet1.20100123.4b5b1954161b21.41424525

worksheet1.20100123.4b5b1954161b21.41424525 - Worksheet on...

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Worksheet on polar coordinates Exploring invertibility Using Taylor series Suppose that there exists values x 1 * , y 1 * , Ρ * , and Φ * such that x 1 * = Ρ * cos H Φ * L and y 1 * = Ρ * sin H Φ * L . Consider two smooth functions Ρ ( x 1 , y 1 ) and Φ ( x 1 , y 1 ) for x 1 » x 1 * and y 1 » y 1 * , such that Ρ ( x 1 * , y 1 * )= Ρ * and Φ ( x 1 * , y 1 * )= Φ * . It follows that there exists numbers Ρ 1 * , Ρ 2 * , Φ 1 * , and Φ 2 * , such that Ρ ( x 1 , y 1 )= Ρ * + Ρ 1 * ( x 1 - x 1 * )+ Ρ 2 * ( y 1 - y 1 * )+ O (2) and Φ ( x 1 , y 1 )= Φ * + Φ 1 * ( x 1 - x 1 * )+ Φ 2 * ( y 1 - y 1 * )+ O (2). Now suppose that Ρ ( x 1 , y 1 ) and Φ ( x 1 , y 1 ) satisfy the equations x 1 = Ρ ( x 1 , y 1 )cos( Φ ( x 1 , y 1 )) and y 1 = Ρ ( x 1 , y 1 )sin( Φ ( x 1 , y 1 )). This is certainly true for x 1 = x 1 * and y 1 = y 1 * . Suppose it is also true for all x 1 » x 1 * and all y 1 » y 1 * . Let Ε Δ x 1 = x 1 - x 1 * and Ε Δ y 1 = y 1 - y 1 * and assume that Ε 1. Then In[1]:= Series @8 Ρ s Cos @ Φ s D + Ε Δ x1 H Ρ s + Ρ 1s Ε Δ x1 + Ρ 2s Ε Δ y1 L Cos @ Φ s + Φ 1s Ε Δ x1 + Φ 2s Ε Δ y1 D , Ρ s Sin @ Φ s D + Ε Δ y1 H Ρ s + Ρ 1s Ε Δ x1 + Ρ 2s Ε Δ y1 L Sin @ Φ s + Φ 1s Ε Δ x1 + Φ 2s Ε Δ
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  • Spring '08
  • Weaver
  • Derivative, Continuous function, Y1 Y1, Cos Φs

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