# If x= rcosФ , y=rsinФ then find dx/x² + dy/y²

1
by LimnaButala

2015-09-24T17:50:16+05:30

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I will us e the symbol  A in place of Ф.

x = r cos A  and y = r sin A

dx = - r sin A  dA + Cos A  dr

dx / x² = [- r Sin A  dA + Cos A  dr ] / (r² cos²A)
dy / y² = [ r cos A dA  + sin A dr  ] / (r² Sin² A)

dx / x²  + dy /y²
=  [ - r Sin³ A dA + Cos A Sin² A dr + r Cos³A dA + Sin A Cos²A dr]
/ (r² Sin²A Cos²A)
= [ r (CosA - SinA) (Cos²A+CosA SinA + Sin²A) dA + (CosA Sin²A + SinA Cos²A) dr ]  / [r² Sin²A Cos²A ]

= [ r (cosA - Sin A)(1+cosA SinA) dA + CosA SinA (sinA + CosA) dr ]
/ [r² Sin²A Cos²A]

= [ (x - y) (1 + xy/r²) dA +  xy/r³ (x + y) dr ] / x²y²

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